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Math / Contemporary Abstract Algebra 8 / Chapter 0 / Problem 10E

Contemporary Abstract Algebra | 8th Edition | ISBN: 9781133599708 | Authors: Joseph Gallian

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Let a and b be positive integers and let d = gcd(a, b) and

Chapter 0, Problem 10E

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QUESTION:

Let a and b be positive integers and let d = gcd(a, b) and m = lcm(a, b). If t divides both a and b, prove that t divides d. If s is a multiple of both a and b, prove that s is a multiple of m.

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QUESTION:

Let a and b be positive integers and let d = gcd(a, b) and m = lcm(a, b). If t divides both a and b, prove that t divides d. If s is a multiple of both a and b, prove that s is a multiple of m.

ANSWER:

Step 1 of 2

Let and, therefore it follows:

for some and .

Let is assume that there is a number which is a factor of both and so it follows:

Therefore it follows:

So divides .

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