Problem

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Tags: modular arithmetic, number theory unsolved, number theory



Let $v(n)$ be the order of $2$ in $n!$. Prove that for any positive integers $a$ and $m$ there exists $n$ ($n>1$) such that $v(n) \equiv a (\mod m)$. I have a book with Mongolian problems from this year, and this problem appeared in it. Perhaps I am terribly misinterpreting this problem, but it seems like it is wrong. Any ideas?