Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 336 — #344
336
Chapter 9
Number Theory
To see how to update the coefficients when at least one of a and b is odd and
ua C vb is even, show that either u and v are both even, or else u b and v C a
are both even.
Problem 9.18.
For any set A of integers,
gcd.A/ WWD the greatest common divisor of the elements of A.
The following useful property of gcd’s of sets is easy to take for granted:
Theorem.
for all finite sets A; B Z.
gcd.A [ B/ D gcd.gcd.A/; gcd.B//;
(AuB)
The theorem has an easy proof as a Corollary of the Unique Factorization Theorem.
In this problem we develop a proof by induction of Theorem (AuB) just
making repeated use of Lemma 9.2.6.b :
.d j a AND d j b/ IFF d j gcd.a; b/: (gcddiv)
The key to proving (AuB) will be generalizing (gcddiv) to finite sets.
Definition. For any subset A Z,
d j A WWD 8a 2 A: d j a:
Lemma.
d j A IFF d j gcd.A/:
for all d 2 Z and finite sets A Z.
(divdef)
(dAdgA)
(a) Prove that
gcd.a; gcd.b; c// D gcd.gcd.a; b/; c/
(gcd-associativity)
for all integers a; b; c.
From here on we write “a [ A” as an abbreviation for “fag [ A.”
(b) Prove that
d j .a [ b [ C / IFF d j .gcd.a; b/ [ C /
for all a; b; d 2 Z, and C Z.
(abCgcd)