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Mongolia TST 2011 Test 1 #1

Source:

November 7, 2011
modular arithmeticnumber theory unsolvednumber theory

Problem Statement

Let v(n)v(n) be the order of 22 in n!n!. Prove that for any positive integers aa and mm there exists nn (n>1n>1) such that v(n)≡a(mod  m)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?
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