   Chapter 2.4, Problem 10E

Chapter
Section
Textbook Problem

Let a be a nonzero integer and b a positive integer. Prove or disprove that ( a ,   b ) = ( a ,   a + b ) .

To determine

Whether (a,b)=(a,a+b), where a is a non-zero integer and b is a positive integer.

Explanation

Given information:

a is a non-zero integer and b is a positive integer.

Formula used:

Greatest common divisor:

Let d=(a,b), then d satisfies the following conditions

1. d is a positive integer.

2. d|a and d|b.

3. c|a and c|b imply c|d.

Calculation:

Let a is a non-zero integer and b is a positive integer and d=(a,b) is greatest common divisor of a and b; therefore, by definition of d|a and d|b.

As d|a and d|b; therefore, there exist integers m and n such that a=dm and b=dn.

Therefore, a+b=dm+dna+b=d(m+n)

Hence, d|(a+b)

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