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Prove that the greatest common divisors are equal

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September 22, 2010

least common multiplegreatest common divisornumber theory

Problem Statement

Let

a_i, b_i

be coprime positive integers for

i = 1, 2, \ldots , k

, and

m

the least common multiple of

b_1, \ldots , b_k

. Prove that the greatest common divisor of

a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}

equals the greatest common divisor of

a_1, \ldots , a_k.