   Chapter 2.4, Problem 22E

Chapter
Section
Textbook Problem

Let ( a ,   b ) = 1 . Prove ( a ,   b c ) = ( a ,   c ) , where c is any integer.

To determine

To prove: If (a,b)=1 then (a,bc)=(a,c), where c is any integer.

Explanation

Given information:

(a,b)=1.

Formula used:

Greatest Common Divisors:

Let a and b be the integers, at least one of them is non-zero. Then, there exists a unique greatest common divisor d of a and b. Moreover, d can be written as d=am+bn for integers m and n, and d is the smallest positive integer that can be written in this form.

Proof:

Let (a,b)=1.

Assume, (a,bc)=d and (a,c)=g.

To show: (a,bc)=(a,c), that is, d=g.

Since, (a,c)=g.

g|a and g|c

g|bc

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