   Chapter 8.4, Problem 33ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Use Theorem 8.4.5to prove that for all integers a, b, and c, if gcd ( a , b ) = 1 and a | c and b | c , then a b | c .

To determine

To prove:

For all integers a,b and c, if gcd(a,b)=1 and a|c and b|c, then ab|c.

Explanation

Given information:

a,b,c are integers.

gcd(a,b)=1

Concept used:

gcd:Greatest common divisor

Proof:

Objective is to prove that if gcd(a,b)=1 and a|c and b|c then ab|c for all integers a,b and c.

For all integers a,b and c,gcd(a,b)=1 and a|c and b|c.

By Euclidean algorithm, there exist integers s,t such that 1=as+bt............(1).

Given that a|cc=ax and b|cc=by for some x,y non-zero integers

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