   Chapter 2.4, Problem 25E

Chapter
Section
Textbook Problem

Prove that if m > 0 and ( a ,   b ) exists, then ( m a ,   m b ) = m   ·   ( a ,   b ) .

To determine

To prove: If m>0 and (a,b) exists, then (ma,mb)=m(a,b).

Explanation

Given information:

m>0 and (a,b) exists.

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 m>0 and (a,b)=c.

Assume, (ma,mb)=d.

To prove: (ma,mb)=m(a,b), that is d=mc.

As, c=(a,b) c|a and c|b

Since, m>0.

mc|ma and mc|mb.

Therefore, mc|(ma,mb)mc|d

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