Skip to Main Content
The Nuts and Bolts of Proofs, 4th Edition
book

The Nuts and Bolts of Proofs, 4th Edition

by Antonella Cupillari
January 2011
Beginner content levelBeginner
296 pages
11h 43m
English
Academic Press
Content preview from The Nuts and Bolts of Proofs, 4th Edition
64. Let a, b, and c be three counting numbers. Set d = GCD(ac, bc) and e = GCD(a, b).
We want to prove that d = ce.
Part 1. d ce
Because e = GCD(a, b), we can write a = ke and b = se with k an d s relatively prime. Multiplying
both equalities by c, we obtain
ac = kðceÞ and bc = sðceÞ:
This proves that ce is a common divisor of ac and bc.Butd is the greatest common divisor.
Thus, d ce.
Part 2. d ce
As e is the greatest common divisor of a and b, we can write a = ke and b = se ,w
ithk and s
relatively prime. So, multiplying by c, we obtain
ac = kðceÞ and bc = sðceÞ,
with k and s relatively prime. Then all the common factors of ac and bc are in ce.T
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Discrete Structures, Logic, and Computability, 4th Edition

Discrete Structures, Logic, and Computability, 4th Edition

James L. Hein

Publisher Resources

ISBN: 9780123822178