   Chapter 2.4, Problem 14E

Chapter
Section
Textbook Problem

If b > 0 and a = b q + r , prove that ( a ,   b ) = ( b ,   r ) .

To determine

To prove: If b>0 and a=bq+r, then (a,b)=(b,r).

Explanation

Given information:

b>0 and a=bq+r

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:

As b>0 and a=bq+r.

Let (a,b)=d and (b,r)=g.

As (a,b)=dd|a,d|b.

Therefore, d|bq.

Hence, d|(abq)d|r

As (b,r)=g, thus, d|g

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