CN-TST 2016 Problem 10

Set positive integer $m=2^k\cdot t$ , where $k$ is a non-negative integer, $t$ is an odd number, and let $f(m)=t^{1-k}$ . Prove that for any positive integer $n$ and for any positive odd number $a\le n$ , $\prod_{m=1}^n f(m)$ is a multiple of $a$ .