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Product of f(m) multiple of odd integers

Source: China Team Selection Test 2016 Test 2 Day 2 Q4

March 21, 2016
number theoryfloor functionHi

Problem Statement

Set positive integer m=2k⋅tm=2^k\cdot t, where kk is a non-negative integer, tt is an odd number, and let f(m)=t1−kf(m)=t^{1-k}. Prove that for any positive integer nn and for any positive odd number a≤na\le n, ∏m=1nf(m)\prod_{m=1}^n f(m) is a multiple of aa.
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