3.3.1
Section 3 - Divisibility
• Definition: If n and d are integers and d ≠ 0, then
n is divisible by d provided n = d ⋅ k for some
integer k.
• Alternatively, we say:
n is a multiple of d
d is a factor of n
d is a divisor of n
d divides n (denoted with d | n).
3.3.2
Properties of Divisibility
• Divisors of 0: If k is a non-zero integer, then
k divides 0 since 0 = k ⋅ 0.
• Positive Divisors of a Positive Number:
If a and b are positive integers and a | b, is a ≤ b?
Yes. Since a | b, ∃ k ∈ Z,such that b = a ⋅ k.
Moreover, 0 < k, since a and b are, so 1 ≤ k.
Thus:
a = a ⋅ 1 ≤ a ⋅ k = b.
Therefore
a ≤ b.
• Divisors of 1: The only divisors of 1 are 1 and −1.
3.3.3
Divisibility of Algebraic Terms
• Let a and b be integers.
• Does 3 | (3a + 3b)?
Yes, since (3a + 3b) = 3(a + b) and (a + b) ∈ Z.
• Does 5 | 10ab?
Yes again, since 10ab = 5(2ab) and (2ab) ∈ Z.
• If m ∈ Z and m | (a + b), does m | a and m | b?
No. 2 | 8 but 2 | 5 and 2 | 3.
3.3.4
Divisibility and Non-divisibility
• There is another way to test for divisibility:
If d | n, there is integer k with n = dk, then
k = (n/d). So, if (n/d) is an integer, then d | n.
• This leads to an easy way to test for nondivisibility: If (n/d) is not an integer, then d
cannot divide n.
• Examples:
3 | 12 since 12/3 = 4 ∈ Z.
5 | 12 since 12/5 = 2.4 ∉ Z.
3.3.5
Proving Properties of Divisibility
• Theorem: Transitivity of Divisibility
For all a,b,c ∈ Z, if a | b and b | c, then a | c.
• Proof: Let a, b, and c be integers, and assume
a | b and b | c. Thus there exist m,n ∈ Z with
b = ma and c = nb.
Now, c = nb = n(ma) = (nm)a. Since
m,n ∈ Z, we have nm ∈ Z, therefore a | c. QED
• Example: 3 | 9 and 9 | 909, therefore 3 | 909.
3.3.6
Divisibility by a Prime
• Theorem: Every positive integer greater than 1
is divisible by a prime number.
• Proof: Let n ∈ Z with n > 1. Then either n is
prime or composite. If n is prime, it is divisible
by itself, and we are done.
Now, assume n is composite. Thus there are
integers (greater than 1) a and b, such that
n = ab. If a is prime, we are done. If not, factor
a, .... Will we eventually get to a prime factor?
3.3.7
Standard Factored Form
• Definition: Given any integer n > 1, the
standard factored form of n is an expression of
the form:
n = (p1)e1 ⋅ (p2)e2 ⋅ (p3)e3...(pk)ek,
where k is a positive integer ; p1,p2,...,pk are
prime numbers with p1 < p2 < ... < pk; and
e1,e2,...,ek are positive integers.
• Example:
3300 = 33 ⋅ 100 = 3 ⋅ 11 ⋅ 102
= 22 ⋅ 3 ⋅ 52 ⋅ 11.
3.3.8
Unique Factorization Theorem
• Theorem: Given any integer n > 1, there exist
positive integer k; prime numbers p1,p2,...,pk;
and positive integers e1,e2,...,ek, with
n = (p1)e1 ⋅ (p2)e2 ⋅ (p3)e3...(pk)ek,
and any other expression of n as a product of
prime numbers is identical to this except,
perhaps, for the order in which the factors
appears.
• This is also referred to as the Fundamental
Theorem of Arithmetic.
Fundamental Theorem
of Arithmetic
3.3.9
• Theorem: Every positive integer greater than 1
has a unique factorization as the product of
primes.
• Proof: (outline)
1. Apply the previous theorem to each composite
factor encountered.
2. Sort the final listing to get the prime factors in
increasing (decreasing?) numeric order.
3. Rewrite using exponents.