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Chapter 2 The Integers
Then all repeated factors may be collected together and expressed by use of exponents to
yield
n 5 p1m1p2m2 c pmr r
where each mi is a positive integer. Each mi is called the multiplicity of pi, and this factorization is known as the standard form for n.
Example 2
The standard forms for two positive integers a and b can be used to find
their greatest common divisor (a, b) and their least common multiple (see Exercises 28 and
29 at the end of this section). For instance, if
a 5 31,752 5 23 ? 34 ? 72
and b 5 126,000 5 24 ? 32 ? 53 ? 7,
then (a, b) can be found by forming the product of all the common prime factors, with each
common factor raised to the least power to which it appears in either factorization:
(a, b) 5 23 ? 32 ? 7 5 504.
■
From one point of view, the Unique Factorization Theorem says that the prime integers are building blocks for the integers, where the “building” is done by using multiplication and forming products. A natural question, then, is: How many blocks? Our
next theorem states the answer given by the ancient Greek mathematician Euclid—that
the number of primes is infinite. The proof is also credited to Euclid.
Theorem 2.19
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Euclid’s Theorem on Primes
The number of primes is infinite.
Contradiction
Proof Assume there are only a finite number, n, of primes. Let these n primes be denoted
by p1, p2, c, pn, and consider the integer
m 5 p1 p2 c pn 1 1.
It is clear that the remainder in the division of m by any prime pi is 1, so each pi is not a factor of m. Thus there are two possibilities: Either m is itself a prime, or it has a prime factor
that is different from every one of the pi. In either case, we have a prime integer that was
not in the list p1, p2, c, pn. Therefore, there are more than n primes, and this contradiction
establishes the theorem.
Exercises 2.4
True or False
Label each of the following statements as either true or false.
1. The set of prime numbers is closed with respect to multiplication.
2. The set of prime numbers is closed with respect to addition.
3. The greatest common divisor is a binary operation from Z 2 {0} 3 Z to Z1.
2.4 Prime Factors and Greatest Common Divisor
93
4. The least common multiple is a binary operation from Z 2 {0} 3 Z 2 {0} to Z1.
5. The greatest common divisor is unique, when it exists.
6. Let a and b be integers, not both zero, such that 1 5 (a, b). Then there exist integers x
and y such that 1 5 ax 1 by and (x, y) 5 1.
7. Let a and b be integers, not both zero, such that d 5 ax 1 by for integers x and y. Then
d 5 (a, b).
8. Let a and b be integers, not both zero, such that d 5 (a, b). Then there exist unique
integers x and y such that d 5 ax 1 by.
9. Let a and b be integers, not both zero. Then (a, b) 5 (2a, b).
10. Let a be an integer, then (a, a 1 1) 5 1.
11. Let a be an integer, then (a, a 1 2) 5 2.
12. If (a, b) 5 1 and (a, c) 5 1, then (b, c) 5 1.
Exercises
In this set of exercises, all variables represent integers.
1. List all the primes less than 100.
2. For each of the following pairs, write a and b in standard form and use these factorizations to find (a, b).
a. a 5 1400, b 5 980
b. a 5 4950, b 5 10,500
c. a 5 3780, b 5 16,200
d. a 5 52,920, b 5 25,200
3. In each part, find the greatest common divisor (a, b) and integers m and n such that
(a, b) 5 am 1 bn.
a. a 5 0, b 5 23
b. a 5 65, b 5 291
c. a 5 102, b 5 66
d. a 5 52, b 5 124
e. a 5 414, b 5 233
f. a 5 252, b 5 2180
g. a 5 414, b 5 693
h. a 5 382, b 5 26
i. a 5 1197, b 5 312
j. a 5 3780, b 5 1200
k. a 5 6420, b 5 132
l. a 5 602, b 5 252
m. a 5 5088, b 5 2156
n. a 5 8767, b 5 252
4. Find the smallest integer in the given set.
a. {x [ Z 0 x . 0 and x 5 4s 1 6t for some s, t in Z}
b. {x [ Z 0 x . 0 and x 5 6s 1 15t for some s, t in Z}
5. Prove that if p and q are distinct primes, then there exist integers m and n such that
pm 1 qn 5 1.
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Chapter 2 The Integers
6. Show that n2 2 n 1 5 is a prime integer when n 5 1, 2, 3, 4 but that it is not true that
n2 2 n 1 5 is always a prime integer. Write out a similar set of statements for the
polynomial n2 2 n 1 11.
7. If a . 0 and a 0 b, then prove or disprove that (a, b) 5 a.
8. Let a, b, and c be integers such that a Z 0. Prove that if a 0 bc, then a 0 c ? (a, b).
9. Let a be a nonzero integer and b a positive integer. Prove or disprove that (a, b) 5
(a, a 1 b).
10. Let a 0 c and b 0 c, and (a, b) 5 1, prove that ab divides c.
11. Prove that if d 5 (a, b), a 0 c, and b 0 c, then ab 0 cd.
12. If b . 0 and a 5 bq 1 r, prove that (a, b) 5 (b, r).
13. Let r0 5 b . 0. With the notation used in the description of the Euclidean
Algorithm, use the result in Exercise 12 to prove that (a, b) 5 rn , the last nonzero
remainder.
14. Prove that every remainder rj in the Euclidean Algorithm is a “linear combination” of
a and b: rj 5 sj a 1 tj b, for integers sj and tj .
15. Let a and b be integers, at least one of them not 0. Prove that an integer c can be
expressed as a linear combination of a and b if and only if (a, b) 0 c.
16. Prove Corollary 2.17: If p is a prime and p 0 (a1a2 p an), then p divides some aj. (Hint:
Use induction on n.)
17. Prove that if n is a positive integer greater than 1 such that n is not a prime, then n has
a divisor d such that 1 , d # !n.
18. Prove that (ab, c) 5 1 if and only if (a, c) 5 1 and (b, c) 5 1.
19. Let (a, b) 5 1 and (a, c) 5 1. Prove or disprove that (ac, b) 5 1.
20. Let (a, b) 5 1. Prove (a, bc) 5 (a, c), where c is any integer.
21. Let (a, b) 5 1. Prove (a2, b2) 5 1.
22. Let (a, b) 5 1. Prove that (a, bn) 5 1 for all positive integers n.
23. Prove that if m . 0 and (a, b) exists, then (ma, mb) 5 m ? (a, b).
Sec. 2.5, #28 !
Sec. 8.2, #35 !
24. Prove that if d 5 (a, b), a 5 a0 d, and b 5 b0d, then (a0, b0) 5 1.
25. A least common multiple of two nonzero integers a and b is an integer m that satisfies
all the following conditions:
1. m is a positive integer.
2. a 0 m and b 0 m.
3. a 0 c and b 0 c imply m 0 c.
Prove that the least common multiple of two nonzero integers exists and is unique.
26. Let a and b be positive integers. If d 5 (a, b) and m is the least common multiple of a
and b, prove that dm 5 ab. Note that it follows that the least common multiple of two
positive relatively prime integers is their product.
2.5 Congruence of Integers
95
27. Let a and b be positive integers. Prove that if d 5 (a, b), a 5 a0d, and b 5 b0d, then
the least common multiple of a and b is a0b0d.
28. Describe a procedure for using the standard forms of two positive integers to find their
least common multiple.
29. For each pair of integers a, b in Exercise 2, find the least common multiple of a and b
by using their standard forms.
30. Let a, b, and c be three nonzero integers.
a. Use Definition 2.11 as a pattern to define a greatest common divisor of a, b, and c.
b. Use Theorem 2.12 and its proof as a pattern to prove the existence of a greatest
common divisor of a, b, and c.
c. If d is the greatest common divisor of a, b, and c, show that d 5 ((a, b), c) .
d. Prove ((a, b), c) 5 (a, (b, c)).
31. Find the greatest common divisor of a, b, and c and write it in the form ax 1 by 1 cz
for integers x, y, and z.
a. a 5 14, b 5 28, c 5 35
b. a 5 26, b 5 52, c 5 60
c. a 5 143, b 5 385, c 5 265
d. a 5 60, b 5 284, c 5 105
32. Use the Second Principle of Finite Induction to prove that every positive integer n can
be expressed in the form
n 5 c0 1 c1 ? 3 1 c2 ? 32 1 c 1 cj21 ? 3 j21 1 cj ? 3 j,
where j is a nonnegative integer, ci [ {0, 1, 2} for all i , j, and cj [ {1, 2}.
33. Use the fact that 2 is a prime to prove that there do not exist nonzero integers a and b
such that a 2 5 2b 2 . Explain how this proves that !2 is not a rational number.
2.5
34. Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such
that a 2 5 3b 2 . Explain how this proves that !3 is not a rational number.
Congruence of Integers
In Example 4 of Section 1.7, we defined the relation “congruence modulo 4” on the set Z
of all integers, and we proved this relation to be an equivalence relation on Z. That example is a special case of congruence modulo n, as defined next.
Definition 2.20 ■ Congruence Modulo n
Let n be a positive integer, n . 1. For integers x and y, x is congruent to y modulo n if and
only if x 2 y is a multiple of n. We write
x ; y (mod n)
to indicate that x is congruent to y modulo n.