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Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230

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Section
BuyFindarrow_forward

Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
ISBN: 9781285463230
Textbook Problem

Prove that if d = ( a , b ) , a = a 0 d , and b = b 0 d , then ( a 0 , b 0 ) = 1 .

To determine

To prove: If d=(a,b),a=a0d and b=b0d, then (a0,b0)=1.

Explanation

Given information:

d=(a,b),a=a0d and b=b0d.

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:

Using definition of greatest common divisors, d=(a,b)ax+by=d for integers x and y.

Given that a=a0d and b=b0d

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