I just encountered this problem from Turkey's team selection test:
$f(a,n)$ is the number of 10-tuples $(x_1,x_2, \cdots, x_{10})$ mod $n$ that satisfies the equation $$x_1x_2...x_{10} \equiv a \textrm{ mod } n.$$ Given $a,b \in \mathbb{N}$, prove that there exists a number $c$ such that $\frac{f(a,cn)}{f(b,cn)}$ is constant.
There is a question I wanted to ask: Is there a generating function that computes $f(a,n)$? If not then what are the characteristics of this function? Edit: I have found this equality for smaller tuples but has yet to prove the generalized version: $f(a,mn)=f(a,m)*f(a,n)$, for all $m, n \in \mathbb{N}$ such that $gcd(m,n)=1$.