Prime Factorization
Least Common Multiples
Roots
The Fundamental Theorem of Arithmetic
Bernd Schröder
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Introduction
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Introduction
1. Prime numbers are indivisible.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Introduction
1. Prime numbers are indivisible.
2. So it stands to reason that everything else is made up of
prime numbers.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Introduction
1. Prime numbers are indivisible.
2. So it stands to reason that everything else is made up of
prime numbers.
3. Proving this very reasonable idea is surprisingly
complicated.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Introduction
1. Prime numbers are indivisible.
2. So it stands to reason that everything else is made up of
prime numbers.
3. Proving this very reasonable idea is surprisingly
complicated.
4. We will also talk about least common multiples and
greatest common divisors.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Introduction
1. Prime numbers are indivisible.
2. So it stands to reason that everything else is made up of
prime numbers.
3. Proving this very reasonable idea is surprisingly
complicated.
4. We will also talk about least common multiples and
greatest common divisors. (For which we can prove that
they exist.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Introduction
1. Prime numbers are indivisible.
2. So it stands to reason that everything else is made up of
prime numbers.
3. Proving this very reasonable idea is surprisingly
complicated.
4. We will also talk about least common multiples and
greatest common divisors. (For which we can prove that
they exist.)
5. And we will apply the new results about division to roots
of polynomials with integer coefficients and leading
coefficient 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n. The base case n = 2 is trivial,
because 2|2.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n. The base case n = 2 is trivial,
because 2|2.
For the induction step, let n > 2 and assume the result holds for
all positive integers k with 2 ≤ k < n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n. The base case n = 2 is trivial,
because 2|2.
For the induction step, let n > 2 and assume the result holds for
all positive integers k with 2 ≤ k < n. If n is prime, choose
p := n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n. The base case n = 2 is trivial,
because 2|2.
For the induction step, let n > 2 and assume the result holds for
all positive integers k with 2 ≤ k < n. If n is prime, choose
p := n. If n is not prime, then there are integers k and m with
2 ≤ k, m < n so that n = km.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n. The base case n = 2 is trivial,
because 2|2.
For the induction step, let n > 2 and assume the result holds for
all positive integers k with 2 ≤ k < n. If n is prime, choose
p := n. If n is not prime, then there are integers k and m with
2 ≤ k, m < n so that n = km. By induction hypothesis, there is a
prime number p so that p|k.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n. The base case n = 2 is trivial,
because 2|2.
For the induction step, let n > 2 and assume the result holds for
all positive integers k with 2 ≤ k < n. If n is prime, choose
p := n. If n is not prime, then there are integers k and m with
2 ≤ k, m < n so that n = km. By induction hypothesis, there is a
prime number p so that p|k. Now p|n, which concludes the
proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let n ≥ 2 be a positive integer. Then there is a prime
number p so that p|n.
Proof. Strong induction on n. The base case n = 2 is trivial,
because 2|2.
For the induction step, let n > 2 and assume the result holds for
all positive integers k with 2 ≤ k < n. If n is prime, choose
p := n. If n is not prime, then there are integers k and m with
2 ≤ k, m < n so that n = km. By induction hypothesis, there is a
prime number p so that p|k. Now p|n, which concludes the
proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p-d
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d. Let a be so that ap = cd.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d. Let a be so that ap = cd.
Then we have the following possibilities
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d. Let a be so that ap = cd.
Then we have the following possibilities
1. a < c ≤ d < p
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d. Let a be so that ap = cd.
Then we have the following possibilities
1. a < c ≤ d < p,
2. p < c ≤ d < a
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d. Let a be so that ap = cd.
Then we have the following possibilities
1. a < c ≤ d < p,
2. p < c ≤ d < a,
3. c < a ≤ p < d
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d. Let a be so that ap = cd.
Then we have the following possibilities
1. a < c ≤ d < p,
2. p < c ≤ d < a,
3. c < a ≤ p < d,
and
4. c < p < a < d.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof. Suppose for a contradiction that the result is false. Then
we can find positive integers c and d so that p|cd and p - c and
p - d and so that c + d is as small as possible.
Without loss of generality, let c ≤ d. Let a be so that ap = cd.
Then we have the following possibilities
1. a < c ≤ d < p,
2. p < c ≤ d < a,
3. c < a ≤ p < d,
and
4. c < p < a < d.
(Any other possibility leads to ap < cd or ap > cd or to a
cancelation of a or p.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap = cd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap = cd = (a + x)(p − y)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap = cd = (a + x)(p − y)
ap = ap − ay + x(p − y)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap = cd = (a + x)(p − y)
ap = ap − ay + x(p − y)
0 = −ay + x(p − y)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap = cd = (a + x)(p − y)
ap = ap − ay + x(p − y)
0 = −ay + x(p − y) = −ay + xd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
Bernd Schröder
The Fundamental Theorem of Arithmetic
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup = ap
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup = ap = cd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup = ap = cd = qvd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup = ap = cd = qvd, so up = vd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup = ap = cd = qvd, so up = vd and p - v
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup = ap = cd = qvd, so up = vd and p - v and p - d
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case a < c ≤ d < p, let x, y be so that c = a + x
(note that x < c) and let y be so that d = p − y. Then
ap
ap
0
ay
=
=
=
=
cd = (a + x)(p − y)
ap − ay + x(p − y)
−ay + x(p − y) = −ay + xd
xd
Let q be a prime factor of a and a = qu. Because c + d was as
small as possible, q|x (so q|c) or q|d, say, c = qv. Then
qup = ap = cd = qvd, so up = vd and p - v and p - d,
contradicting that c + d was as small as possible.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap = cd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap = cd = (p + x)(a − y)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap = cd = (p + x)(a − y)
ap = pa − py + x(a − y)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap = cd = (p + x)(a − y)
ap = pa − py + x(a − y)
0 = −py + x(a − y)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap = cd = (p + x)(a − y)
ap = pa − py + x(a − y)
0 = −py + x(a − y) = −py + xd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap
ap
0
py
Bernd Schröder
The Fundamental Theorem of Arithmetic
=
=
=
=
cd = (p + x)(a − y)
pa − py + x(a − y)
−py + x(a − y) = −py + xd
xd
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap
ap
0
py
=
=
=
=
cd = (p + x)(a − y)
pa − py + x(a − y)
−py + x(a − y) = −py + xd
xd
Now, because c + d was as small as possible, p|x or p|d.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap
ap
0
py
=
=
=
=
cd = (p + x)(a − y)
pa − py + x(a − y)
−py + x(a − y) = −py + xd
xd
Now, because c + d was as small as possible, p|x or p|d.
Because p - d, we must have p|x.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case p < c ≤ d < a, let x, y be so that c = p + x
(note that x < c) and let y be so that d = a − y. Then
ap
ap
0
py
=
=
=
=
cd = (p + x)(a − y)
pa − py + x(a − y)
−py + x(a − y) = −py + xd
xd
Now, because c + d was as small as possible, p|x or p|d.
Because p - d, we must have p|x. However this implies that p|c,
a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case c < a ≤ p < d, let x, y be so that c = a − x
and let y be so that d = p + y (note that y < d). Then
ap
ap
0
xp
=
=
=
=
cd = (a − x)(p + y)
ap − xp + (a − x)y
−xp + (a − x)y = −xp + cy
cy
Now, because c + d was as small as possible, p|c or p|y.
Because p - c, we must have p|y. However, this implies that p|d,
a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case c < p < a < d, let x, y be so that c = a − x
and let y be so that d = p + y (note that y < d). Then
ap
ap
0
xp
=
=
=
=
cd = (a − x)(p + y)
ap − xp + (a − x)y
−xp + (a − x)y = −xp + cy
cy
Now, because c + d was as small as possible, p|c or p|y.
Because p - c, we must have p|y. However, this implies that p|d,
a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let c, d be positive integers and let p be a prime
number. If p|cd, then p|c or p|d.
Proof (cont.) In case c < p < a < d, let x, y be so that c = a − x
and let y be so that d = p + y (note that y < d). Then
ap
ap
0
xp
=
=
=
=
cd = (a − x)(p + y)
ap − xp + (a − x)y
−xp + (a − x)y = −xp + cy
cy
Now, because c + d was as small as possible, p|c or p|y.
Because p - c, we must have p|y. However, this implies that p|d,
a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2. Follows from the previous lemma.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2. Follows from the previous lemma.
Induction step, n → n + 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2. Follows from the previous lemma.
Induction step, n → n + 1. Let p|a1 · · · an an+1 .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2. Follows from the previous lemma.
Induction step, n → n + 1. Let p|a1 · · · an an+1 . Then
p|(a1 · · · an )an+1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2. Follows from the previous lemma.
Induction step, n → n + 1. Let p|a1 · · · an an+1 . Then
p|(a1 · · · an )an+1 and by the previous lemma, we have p|a1 · · · an
or p|an+1 .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2. Follows from the previous lemma.
Induction step, n → n + 1. Let p|a1 · · · an an+1 . Then
p|(a1 · · · an )an+1 and by the previous lemma, we have p|a1 · · · an
or p|an+1 . By induction hypothesis, p|aj for some
j ∈ {1, . . . , n + 1}.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let a1 , . . . , an be positive integers and let p be prime.
If p|a1 · · · an , then there is an i so that p|ai .
Proof. The proof is an induction on n.
Base step, n = 2. Follows from the previous lemma.
Induction step, n → n + 1. Let p|a1 · · · an an+1 . Then
p|(a1 · · · an )an+1 and by the previous lemma, we have p|a1 · · · an
or p|an+1 . By induction hypothesis, p|aj for some
j ∈ {1, . . . , n + 1}.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Existence and uniqueness of a prime
factorization/Fundamental Theorem of Arithmetic/Unique
Factorization Theorem.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Existence and uniqueness of a prime
factorization/Fundamental Theorem of Arithmetic/Unique
Factorization Theorem. Let n ∈ N.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Existence and uniqueness of a prime
factorization/Fundamental Theorem of Arithmetic/Unique
Factorization Theorem. Let n ∈ N. Then either n = 1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Existence and uniqueness of a prime
factorization/Fundamental Theorem of Arithmetic/Unique
Factorization Theorem. Let n ∈ N. Then either n = 1 or there
are unique distinct prime numbers p1 , . . . , pk and (not
necessarily distinct) exponents q1 , . . . , qk ∈ N
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Existence and uniqueness of a prime
factorization/Fundamental Theorem of Arithmetic/Unique
Factorization Theorem. Let n ∈ N. Then either n = 1 or there
are unique distinct prime numbers p1 , . . . , pk and (not
k
q
necessarily distinct) exponents q1 , . . . , qk ∈ N so that n = ∏ pj j .
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence).
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N. If n is prime, choose k = 1, p1 = n,
q1 = 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N. If n is prime, choose k = 1, p1 = n,
q1 = 1. If n is composite, then, by our first lemma, there is a
prime number p so that n = mp with m ∈ {2, . . . , n − 1}.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N. If n is prime, choose k = 1, p1 = n,
q1 = 1. If n is composite, then, by our first lemma, there is a
prime number p so that n = mp with m ∈ {2, . . . , n − 1}. By
induction hypothesis, there are pairwise distinct prime numbers
k
q
p1 , . . . , pk and exponents q1 , . . . , qk ∈ N so that m = ∏ pj j .
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N. If n is prime, choose k = 1, p1 = n,
q1 = 1. If n is composite, then, by our first lemma, there is a
prime number p so that n = mp with m ∈ {2, . . . , n − 1}. By
induction hypothesis, there are pairwise distinct prime numbers
k
q
p1 , . . . , pk and exponents q1 , . . . , qk ∈ N so that m = ∏ pj j .
j=1
k
q
Hence n = p ∏ pj j .
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N. If n is prime, choose k = 1, p1 = n,
q1 = 1. If n is composite, then, by our first lemma, there is a
prime number p so that n = mp with m ∈ {2, . . . , n − 1}. By
induction hypothesis, there are pairwise distinct prime numbers
k
q
p1 , . . . , pk and exponents q1 , . . . , qk ∈ N so that m = ∏ pj j .
j=1
k
q
Hence n = p ∏ pj j . The extra factor p is either equal to some
j=1
pj , in which case we add 1 to the exponent qj
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (existence). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N. If n is prime, choose k = 1, p1 = n,
q1 = 1. If n is composite, then, by our first lemma, there is a
prime number p so that n = mp with m ∈ {2, . . . , n − 1}. By
induction hypothesis, there are pairwise distinct prime numbers
k
q
p1 , . . . , pk and exponents q1 , . . . , qk ∈ N so that m = ∏ pj j .
j=1
k
q
Hence n = p ∏ pj j . The extra factor p is either equal to some
j=1
pj , in which case we add 1 to the exponent qj , or it is not, in
which case we set pk+1 := p and qk+1 := 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness).
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N and assume that uniqueness of the
prime factorization has been proved for all m ∈ {2, . . . , n − 1}.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N and assume that uniqueness of the
prime factorization has been proved for all m ∈ {2, . . . , n − 1}.
If n is a prime number, there is nothing to prove.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N and assume that uniqueness of the
prime factorization has been proved for all m ∈ {2, . . . , n − 1}.
If n is a prime number, there is nothing to prove. So let n be a
composite number and suppose for a contradiction that
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N and assume that uniqueness of the
prime factorization has been proved for all m ∈ {2, . . . , n − 1}.
If n is a prime number, there is nothing to prove. So let n be a
composite number and suppose for a contradiction that there are
pairwise distinct prime numbers p1 , . . . , pk and exponents
k
q
q1 , . . . , qk ∈ N so that n = ∏ pj j as well as pairwise distinct
j=1
prime numbers p̂1 , . . . , p̂k̂ and exponents q̂1 , . . . , q̂k̂ ∈ N so that
k̂
q̂
n = ∏ p̂j j .
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N and assume that uniqueness of the
prime factorization has been proved for all m ∈ {2, . . . , n − 1}.
If n is a prime number, there is nothing to prove. So let n be a
composite number and suppose for a contradiction that there are
pairwise distinct prime numbers p1 , . . . , pk and exponents
k
q
q1 , . . . , qk ∈ N so that n = ∏ pj j as well as pairwise distinct
j=1
prime numbers p̂1 , . . . , p̂k̂ and exponents q̂1 , . . . , q̂k̂ ∈ N so that
k̂
q̂
k̂
q̂
n = ∏ p̂j j . Because p1 divides n = ∏ p̂j j , p1 must divide one
j=1
j=1
of the p̂j .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N and assume that uniqueness of the
prime factorization has been proved for all m ∈ {2, . . . , n − 1}.
If n is a prime number, there is nothing to prove. So let n be a
composite number and suppose for a contradiction that there are
pairwise distinct prime numbers p1 , . . . , pk and exponents
k
q
q1 , . . . , qk ∈ N so that n = ∏ pj j as well as pairwise distinct
j=1
prime numbers p̂1 , . . . , p̂k̂ and exponents q̂1 , . . . , q̂k̂ ∈ N so that
k̂
q̂
k̂
q̂
n = ∏ p̂j j . Because p1 divides n = ∏ p̂j j , p1 must divide one
j=1
j=1
of the p̂j . Assume without loss of generality that p1 = p̂1 .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (uniqueness). Use strong induction on n.
Base step, n = 1. Trivial.
Induction step. Let n ∈ N and assume that uniqueness of the
prime factorization has been proved for all m ∈ {2, . . . , n − 1}.
If n is a prime number, there is nothing to prove. So let n be a
composite number and suppose for a contradiction that there are
pairwise distinct prime numbers p1 , . . . , pk and exponents
k
q
q1 , . . . , qk ∈ N so that n = ∏ pj j as well as pairwise distinct
j=1
prime numbers p̂1 , . . . , p̂k̂ and exponents q̂1 , . . . , q̂k̂ ∈ N so that
k̂
q̂
k̂
q̂
n = ∏ p̂j j . Because p1 divides n = ∏ p̂j j , p1 must divide one
j=1
j=1
of the p̂j . Assume without loss of generality that p1 = p̂1 . Then
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
Roots
q
p̂1 pq11 −1 ∏ pj j =
j=2
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
q
k
Roots
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j
j=2
Bernd Schröder
The Fundamental Theorem of Arithmetic
j=2
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
q
k
q
Roots
k
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j
j=2
Bernd Schröder
The Fundamental Theorem of Arithmetic
j=2
j=1
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
q
k
q
Roots
k
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
Bernd Schröder
The Fundamental Theorem of Arithmetic
j=2
j=1
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
q
Roots
k
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
q̂
∏ p̂j j
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
q
Roots
k
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
The Fundamental Theorem of Arithmetic
k̂
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
j=1
Bernd Schröder
q̂
j=1
j=2
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j
j=2
Bernd Schröder
The Fundamental Theorem of Arithmetic
j=2
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j ∈ {2, . . . , n − 1}.
j=2
Bernd Schröder
The Fundamental Theorem of Arithmetic
j=2
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j ∈ {2, . . . , n − 1}.
j=2
j=2
By induction hypothesis, the two factorizations must be the
same.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j ∈ {2, . . . , n − 1}.
j=2
j=2
By induction hypothesis, the two factorizations must be the
same. So p1 = p̂1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j ∈ {2, . . . , n − 1}.
j=2
j=2
By induction hypothesis, the two factorizations must be the
same. So p1 = p̂1 , q1 = q̂1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j ∈ {2, . . . , n − 1}.
j=2
j=2
By induction hypothesis, the two factorizations must be the
same. So p1 = p̂1 , q1 = q̂1 , k = k̂
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j ∈ {2, . . . , n − 1}.
j=2
j=2
By induction hypothesis, the two factorizations must be the
same. So p1 = p̂1 , q1 = q̂1 , k = k̂ and there is a bijective
function σ : {2, . . . , k} → {2, . . . , k} so that for all j ∈ {2, . . . , k}
we have pj = p̂σ (j) and qj = q̂σ (j) .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
k
k
q
Roots
k
q
q
p̂1 pq11 −1 ∏ pj j = p1 pq11 −1 ∏ pj j = ∏ pj j = n
j=2
j=2
k̂
=
j=1
k̂
q̂
j=1
k
q̂
∏ p̂j j = p̂1p̂q̂11−1 ∏ p̂j j ,
q
j=2
k̂
q̂
which implies that pq11 −1 ∏ pj j = p̂q̂11 −1 ∏ p̂j j ∈ {2, . . . , n − 1}.
j=2
j=2
By induction hypothesis, the two factorizations must be the
same. So p1 = p̂1 , q1 = q̂1 , k = k̂ and there is a bijective
function σ : {2, . . . , k} → {2, . . . , k} so that for all j ∈ {2, . . . , k}
we have pj = p̂σ (j) and qj = q̂σ (j) .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof. Let p be a prime factor of a and let k the exponent with
which p occurs in the prime factorization of a.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof. Let p be a prime factor of a and let k the exponent with
which p occurs in the prime factorization of a. Because
(a, b) = 1, we have p - b.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof. Let p be a prime factor of a and let k the exponent with
which p occurs in the prime factorization of a. Because
(a, b) = 1, we have p - b. Thus p must be a prime factor of c and
it must have an exponent of at least k in the prime factorization
of c
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof. Let p be a prime factor of a and let k the exponent with
which p occurs in the prime factorization of a. Because
(a, b) = 1, we have p - b. Thus p must be a prime factor of c and
it must have an exponent of at least k in the prime factorization
of c (quick induction, repeatedly divide off factors p).
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof. Let p be a prime factor of a and let k the exponent with
which p occurs in the prime factorization of a. Because
(a, b) = 1, we have p - b. Thus p must be a prime factor of c and
it must have an exponent of at least k in the prime factorization
of c (quick induction, repeatedly divide off factors p). Because
p was arbitrary, the whole prime factorization of a must be a
subproduct of the prime factorization of c
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof. Let p be a prime factor of a and let k the exponent with
which p occurs in the prime factorization of a. Because
(a, b) = 1, we have p - b. Thus p must be a prime factor of c and
it must have an exponent of at least k in the prime factorization
of c (quick induction, repeatedly divide off factors p). Because
p was arbitrary, the whole prime factorization of a must be a
subproduct of the prime factorization of c, which means a|c.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Corollary. Let a, b, c be positive integers with (a, b) = 1 and
a|bc. Then a|c.
Proof. Let p be a prime factor of a and let k the exponent with
which p occurs in the prime factorization of a. Because
(a, b) = 1, we have p - b. Thus p must be a prime factor of c and
it must have an exponent of at least k in the prime factorization
of c (quick induction, repeatedly divide off factors p). Because
p was arbitrary, the whole prime factorization of a must be a
subproduct of the prime factorization of c, which means a|c.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Definition.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Definition. Let m, n ∈ Z.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Definition. Let m, n ∈ Z. The greatest common divisor (m, n)
of m and n is the largest d ∈ N so that d|m and d|n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Definition. Let m, n ∈ Z. The greatest common divisor (m, n)
of m and n is the largest d ∈ N so that d|m and d|n. The least
common multiple [m, n] of m and n is the smallest d ∈ N so
that m|d and n|d.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let m, n ∈ N
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let m, n ∈ N and let p1 , . . . , pk be the prime numbers
that occur in the prime factorizations of m and n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let m, n ∈ N and let p1 , . . . , pk be the prime numbers
that occur in the prime factorizations of m and n. Let
k
a
k
b
m = ∏ pj j and n = ∏ pj j
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
j=1
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let m, n ∈ N and let p1 , . . . , pk be the prime numbers
that occur in the prime factorizations of m and n. Let
k
a
k
b
m = ∏ pj j and n = ∏ pj j , where aj and bj could be zero, too.
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
j=1
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let m, n ∈ N and let p1 , . . . , pk be the prime numbers
that occur in the prime factorizations of m and n. Let
k
k
a
b
m = ∏ pj j and n = ∏ pj j , where aj and bj could be zero, too.
j=1
j=1
Then the following hold.
k
min{aj ,bj }
1. (m, n) = ∏ pj
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let m, n ∈ N and let p1 , . . . , pk be the prime numbers
that occur in the prime factorizations of m and n. Let
k
k
a
b
m = ∏ pj j and n = ∏ pj j , where aj and bj could be zero, too.
j=1
j=1
Then the following hold.
k
min{aj ,bj }
1. (m, n) = ∏ pj
j=1
k
max{aj ,bj }
2. [m, n] = ∏ pj
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let m, n ∈ N and let p1 , . . . , pk be the prime numbers
that occur in the prime factorizations of m and n. Let
k
k
a
b
m = ∏ pj j and n = ∏ pj j , where aj and bj could be zero, too.
j=1
j=1
Then the following hold.
k
min{aj ,bj }
1. (m, n) = ∏ pj
j=1
k
max{aj ,bj }
2. [m, n] = ∏ pj
j=1
3. mn = (m, n)[m, n]
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (only parts 1 and 3).
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n. Then the prime
factors of d must be among p1 , . . . , pk .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n. Then the prime
k
c
factors of d must be among p1 , . . . , pk . Hence d = ∏ pj j .
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n. Then the prime
k
c
factors of d must be among p1 , . . . , pk . Hence d = ∏ pj j . If any
j=1
cj was greater than min{aj , bj }
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n. Then the prime
k
c
factors of d must be among p1 , . . . , pk . Hence d = ∏ pj j . If any
j=1
cj was greater than min{aj , bj }, then d would be divisible by a
higher power of pj than m or n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n. Then the prime
k
c
factors of d must be among p1 , . . . , pk . Hence d = ∏ pj j . If any
j=1
cj was greater than min{aj , bj }, then d would be divisible by a
higher power of pj than m or n. Hence for all j ∈ {1, . . . , k} we
have cj ≤ min{aj , bj }.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n. Then the prime
k
c
factors of d must be among p1 , . . . , pk . Hence d = ∏ pj j . If any
j=1
cj was greater than min{aj , bj }, then d would be divisible by a
higher power of pj than m or n. Hence for all j ∈ {1, . . . , k} we
k
have cj ≤ min{aj , bj }. Thus d = ∏
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
c
pj j
k
min{aj ,bj }
≤ ∏ pj
j=1
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
k
min{aj ,bj }
Proof (only parts 1 and 3). For 1, note that ∏ pj
j=1
clearly divides both m and n. Now let d|m, n. Then the prime
k
c
factors of d must be among p1 , . . . , pk . Hence d = ∏ pj j . If any
j=1
cj was greater than min{aj , bj }, then d would be divisible by a
higher power of pj than m or n. Hence for all j ∈ {1, . . . , k} we
k
have cj ≤ min{aj , bj }. Thus d = ∏
j=1
k
min{aj ,bj }
hence (m, n) = ∏ pj
c
pj j
k
min{aj ,bj }
≤ ∏ pj
and
j=1
.
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3).
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3). We need to prove that for all real numbers x, y
max(x, y) + min(x, y) = x + y.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3). We need to prove that for all real numbers x, y
max(x, y) + min(x, y) = x + y.
In case x ≤ y we have the following.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3). We need to prove that for all real numbers x, y
max(x, y) + min(x, y) = x + y.
In case x ≤ y we have the following.
max(x, y) + min(x, y)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3). We need to prove that for all real numbers x, y
max(x, y) + min(x, y) = x + y.
In case x ≤ y we have the following.
max(x, y) + min(x, y) = y
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3). We need to prove that for all real numbers x, y
max(x, y) + min(x, y) = x + y.
In case x ≤ y we have the following.
max(x, y) + min(x, y) = y + x
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3). We need to prove that for all real numbers x, y
max(x, y) + min(x, y) = x + y.
In case x ≤ y we have the following.
max(x, y) + min(x, y) = y + x
The case x > y is similar.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proof (part 3). We need to prove that for all real numbers x, y
max(x, y) + min(x, y) = x + y.
In case x ≤ y we have the following.
max(x, y) + min(x, y) = y + x
The case x > y is similar.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
k
a
Proof. Let d = ∏ pj j be the prime factorization of d.
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
k
a
Proof. Let d = ∏ pj j be the prime factorization of d. (We will
j=1
always assume that the pj in the prime factorization are
distinct.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
k
a
Proof. Let d = ∏ pj j be the prime factorization of d. (We will
j=1
always assume that the pj in the prime factorization are
k
b
distinct.) Let dm be the product dm = ∏ pj j , where bj ≤ aj is the
j=1
b
largest possible exponent so that pj j m
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
k
a
Proof. Let d = ∏ pj j be the prime factorization of d. (We will
j=1
always assume that the pj in the prime factorization are
k
b
distinct.) Let dm be the product dm = ∏ pj j , where bj ≤ aj is the
j=1
k
b
a −b
largest possible exponent so that pj j m and let dn = ∏ pj j j .
j=1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
k
a
Proof. Let d = ∏ pj j be the prime factorization of d. (We will
j=1
always assume that the pj in the prime factorization are
k
b
distinct.) Let dm be the product dm = ∏ pj j , where bj ≤ aj is the
j=1
k
b
a −b
largest possible exponent so that pj j m and let dn = ∏ pj j j .
j=1
The converse is trivial.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let m, n be positive integers and let d be a divisor of
mn. Then there are positive integers dm and dn so that d = dm dn ,
dm |m and dn |n. Conversely, if dm |m and dn |n, then dm dn |mn.
k
a
Proof. Let d = ∏ pj j be the prime factorization of d. (We will
j=1
always assume that the pj in the prime factorization are
k
b
distinct.) Let dm be the product dm = ∏ pj j , where bj ≤ aj is the
j=1
k
b
a −b
largest possible exponent so that pj j m and let dn = ∏ pj j j .
j=1
The converse is trivial.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof. Let a = 4m + 1 and b = 4n + 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof. Let a = 4m + 1 and b = 4n + 1. Then
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof. Let a = 4m + 1 and b = 4n + 1. Then
ab
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof. Let a = 4m + 1 and b = 4n + 1. Then
ab = (4m + 1)(4n + 1)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof. Let a = 4m + 1 and b = 4n + 1. Then
ab = (4m + 1)(4n + 1)
= 16mn + 4m + 4n + 1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof. Let a = 4m + 1 and b = 4n + 1. Then
ab = (4m + 1)(4n + 1)
= 16mn + 4m + 4n + 1
= 4(4mn + m + n) + 1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Lemma. Let a, b be integers of the form 4k + 1. Then ab is of
this form, too.
Proof. Let a = 4m + 1 and b = 4n + 1. Then
ab = (4m + 1)(4n + 1)
= 16mn + 4m + 4n + 1
= 4(4mn + m + n) + 1
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj . Then x is odd
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj . Then x is odd, and one of x + 2 and x + 4 is of the
form 4k + 3.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj . Then x is odd, and one of x + 2 and x + 4 is of the
form 4k + 3. Let y be the number among x + 2 and x + 4 that is
of the form 4k + 3.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj . Then x is odd, and one of x + 2 and x + 4 is of the
form 4k + 3. Let y be the number among x + 2 and x + 4 that is
of the form 4k + 3. Then none of p1 , . . . , pn divides y.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj . Then x is odd, and one of x + 2 and x + 4 is of the
form 4k + 3. Let y be the number among x + 2 and x + 4 that is
of the form 4k + 3. Then none of p1 , . . . , pn divides y. And by
the preceding lemma, some prime of the form 4k + 3 must
divide y (otherwise y = 4m + 1)
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj . Then x is odd, and one of x + 2 and x + 4 is of the
form 4k + 3. Let y be the number among x + 2 and x + 4 that is
of the form 4k + 3. Then none of p1 , . . . , pn divides y. And by
the preceding lemma, some prime of the form 4k + 3 must
divide y (otherwise y = 4m + 1), a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. (Special case of Dirichlet’s Theorem.) There are
infinitely many prime numbers of the form 4k + 3.
Proof. Suppose for a contradiction that there are finitely many
prime numbers of the form 4k + 3. Call them p1 , . . . , pn and let
x := ∏nj=1 pj . Then x is odd, and one of x + 2 and x + 4 is of the
form 4k + 3. Let y be the number among x + 2 and x + 4 that is
of the form 4k + 3. Then none of p1 , . . . , pn divides y. And by
the preceding lemma, some prime of the form 4k + 3 must
divide y (otherwise y = 4m + 1), a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Theorem. If a, b are integers with (a, b) = d, then
Bernd Schröder
The Fundamental Theorem of Arithmetic
Roots
a b
d, d
= 1.
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Theorem. If a, b are integers with (a, b) = d, then
Roots
a b
d, d
= 1.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. If a, b are integers with (a, b) = d, then
Proof. Suppose for a contradiction that
Bernd Schröder
The Fundamental Theorem of Arithmetic
a b
d, d
a b
d, d
= 1.
= x > 1.
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. If a, b are integers with (a, b) = d, then
Proof. Suppose for a contradiction that
a = mdx and b = ndx.
Bernd Schröder
The Fundamental Theorem of Arithmetic
a b
d, d
a b
d, d
= 1.
= x > 1. Then
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Theorem. If a, b are integers with (a, b) = d, then
Roots
a b
d, d
= 1.
Proof. Suppose for a contradiction that da , db = x > 1. Then
a = mdx and b = ndx. But this means that dx > d is a common
divisor of a and b, contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Theorem. If a, b are integers with (a, b) = d, then
Roots
a b
d, d
= 1.
Proof. Suppose for a contradiction that da , db = x > 1. Then
a = mdx and b = ndx. But this means that dx > d is a common
divisor of a and b, contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Theorem. If a, b are integers with (a, b) = d, then
Roots
a b
d, d
= 1.
Proof. Suppose for a contradiction that da , db = x > 1. Then
a = mdx and b = ndx. But this means that dx > d is a common
divisor of a and b, contradiction.
Note that this means that rational numbers can be written “in
lowest terms.”
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 .
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2.
d
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2. WLOG, we can assume that n and d have no
d
common factors and that n ∈ N.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2. WLOG, we can assume that n and d have no
d
common factors and that n ∈ N. But n2 = 2d2 implies n = n2 · 2.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2. WLOG, we can assume that n and d have no
d
common factors and that n ∈ N. But n2 = 2d2 implies n = n2 · 2.
Consequently, 2d2 = (n2 · 2)2
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2. WLOG, we can assume that n and d have no
d
common factors and that n ∈ N. But n2 = 2d2 implies n = n2 · 2.
Consequently, 2d2 = (n2 · 2)2 , that is, d2 = n22 · 2
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2. WLOG, we can assume that n and d have no
d
common factors and that n ∈ N. But n2 = 2d2 implies n = n2 · 2.
Consequently, 2d2 = (n2 · 2)2 , that is, d2 = n22 · 2, which implies
d = d2 · 2.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2. WLOG, we can assume that n and d have no
d
common factors and that n ∈ N. But n2 = 2d2 implies n = n2 · 2.
Consequently, 2d2 = (n2 · 2)2 , that is, d2 = n22 · 2, which implies
d = d2 · 2. But then 2|n and 2|d, contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Proposition. There is no rational number r such that r2 = 2.
Proof. First let n ∈ N be so that n2 = 2z for some z ∈ N. Then 2
is a factor in the prime factorization of n2 . But every factor in
the prime factorization of n2 occurs with an even exponent.
Thus n2 = 22 k2 for some k. Hence n = 2k. That is, if n2 is
divisible by 2, then so is n.
Now suppose for a contradiction that there are n ∈ Z and d ∈ N
n 2
so that
= 2. WLOG, we can assume that n and d have no
d
common factors and that n ∈ N. But n2 = 2d2 implies n = n2 · 2.
Consequently, 2d2 = (n2 · 2)2 , that is, d2 = n22 · 2, which implies
d = d2 · 2. But then 2|n and 2|d, contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Proof.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Proof. Suppose for
a contradiction that there are u ∈ Z and
d ∈ N so that p du = 0, d > 1 and (|u|, d) = 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Proof. Suppose for
a contradiction that there are u ∈ Z and
d ∈ N so that p du = 0, d > 1 and (|u|, d) = 1. Then
u
0 = p
d
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Proof. Suppose for
a contradiction that there are u ∈ Z and
d ∈ N so that p du = 0, d > 1 and (|u|, d) = 1. Then
u
0 = p
d
u n
u n−1
u
=
+ cn−1
+ · · · + c1
+ c0
d
d
d
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Proof. Suppose for
a contradiction that there are u ∈ Z and
d ∈ N so that p du = 0, d > 1 and (|u|, d) = 1. Then
u
0 = p
d
u n
u n−1
u
=
+ cn−1
+ · · · + c1
+ c0
d
d
d
un
0 =
+ cn−1 un−1 + · · · + c1 udn−2 + c0 dn−1
d
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Proof. Suppose for
a contradiction that there are u ∈ Z and
d ∈ N so that p du = 0, d > 1 and (|u|, d) = 1. Then
u
0 = p
d
u n
u n−1
u
=
+ cn−1
+ · · · + c1
+ c0
d
d
d
un
0 =
+ cn−1 un−1 + · · · + c1 udn−2 + c0 dn−1
d
Because all but the first summand are integers, we must have
d|un , which means that (|u|, d) > 1, a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Theorem. Let p(x) = xn + cn−1 xn−1 + · · · + c1 x + c0 be a
polynomial with integer coefficients. If α is a root of p, then α
is either an integer or an irrational number.
Proof. Suppose for
a contradiction that there are u ∈ Z and
d ∈ N so that p du = 0, d > 1 and (|u|, d) = 1. Then
u
0 = p
d
u n
u n−1
u
=
+ cn−1
+ · · · + c1
+ c0
d
d
d
un
0 =
+ cn−1 un−1 + · · · + c1 udn−2 + c0 dn−1
d
Because all but the first summand are integers, we must have
d|un , which means that (|u|, d) > 1, a contradiction.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Example.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Example. In the preceding result, the leading coefficient must
be 1.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Example. In the preceding result, the leading coefficient must
be 1.
To see this, consider p(x) = 9x2 − 4.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Example. In the preceding result, the leading coefficient must
be 1.
To see this, consider p(x) = 9x2 − 4. The coefficients are
integers
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Example. In the preceding result, the leading coefficient must
be 1.
To see this, consider p(x) = 9x2 − 4. The coefficients are
integers and x = 23 is a solution.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science
Prime Factorization
Least Common Multiples
Roots
Example. In the preceding result, the leading coefficient must
be 1.
To see this, consider p(x) = 9x2 − 4. The coefficients are
integers and x = 23 is a solution.
Bernd Schröder
The Fundamental Theorem of Arithmetic
Louisiana Tech University, College of Engineering and Science