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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.

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