Is factorization modulo a product of primes an NP-hard problem? For example, let, $p$ and $q$ be two large prime numbers. We set $n = p \cdot q$.
Now, let $a \cdot b = c \pmod n$.
Given $c$ and $n$, is finding the factors $a$ and $b$ computationally difficult?

I can alternatively ask, is integer factorization modulo a product of primes an NP-hard problem?
Your question has not a unique answer. In general case, finding $a,b$ is very simple. In this state, we can choose an arbitrary $b$(can be a prime number) and find the $a$ as follow:
$$a=c\cdot b^{-1} \pmod n.$$
Unfortunately, if you need that $a$ be prime, your question has no unique answer again. As an example, let $n=33$ and $c=20$. So we have:
$$7\cdot17=23\cdot31=20 \pmod{33}.$$