Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 332 — #340
332
Chapter 9
Number Theory
Problem 9.9. (a) Use the Pulverizer to find gcd.84; 108/
(b) Find integers x, y with 0 y < 84 such that
x 84 C y 108 D gcd.84; 108/:
(c) Is there a multiplicative inverse of 84 in Z 108 ? If not briefly explain why,
otherwise find it.
Problem 9.10.
Indicate true or false for the following statements about the greatest
common divisor, and provide counterexamples for those that are false.
(a) If gcd.a; b/ ¤ 1 and gcd.b; c/ ¤ 1, then gcd.a; c/ ¤ 1. true false
(b) If a j bc and gcd.a; b/ D 1, then a j c. true false
(c) gcd.a n ; b n / D .gcd.a; b// n true false
(d) gcd.ab; ac/ D a gcd.b; c/. true false
(e) gcd.1 C a; 1 C b/ D 1 C gcd.a; b/. true false
(f) If an integer linear combination of a and b equals 1, then so does some integer
linear combination of a and b 2 . true false
(g) If no integer linear combination of a and b equals 2, then neither does any
integer linear combination of a 2 and b 2 . true false
Problem 9.11.
For nonzero integers a, b, prove the following properties of divisibility and GCD’S.
You may use Theorem 9.2.2 that gcd.a; b/ is an integer linear combination of a and
b. You may not appeal to uniqueness of prime factorization Theorem 9.4.1, because
some of these properties are needed to prove unique factorization.)
(a) Every common divisor of a and b divides gcd.a; b/.
(b) gcd.ka; kb/ D k gcd.a; b/ for all k > 0.
(c) If a j bc and gcd.a; b/ D 1, then a j c.
(d) If p j bc for some prime p then p j b or p j c.