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Prove an Integer

Source: China TST 2004 Quiz

February 1, 2009
number theoryprime factorizationnumber theory unsolved

Problem Statement

Let m1 m_1, m2 m_2, ⋯ \cdots, mr m_r (may not distinct) and n1 n_1, n2 n_2 ⋯ \cdots, ns n_s (may not distinct) be two groups of positive integers such that for any positive integer d d larger than 1 1, the numbers of which can be divided by d d in group m1 m_1, m2 m_2, ⋯ \cdots, mr m_r (including repeated numbers) are no less than that in group n1 n_1, n2 n_2 ⋯ \cdots, ns n_s (including repeated numbers). Prove that m1⋅m2⋯mrn1⋅n2⋯ns \displaystyle \frac{m_1 \cdot m_2 \cdots m_r}{n_1 \cdot n_2 \cdots n_s} is integer.
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