Suppose you had a very large composite number, which is the product of a couple large primes. Let's call this $a = c_1c_2c_3c_4$ where $c_1,c_2,c_3$ and $c_4$ are all primes.
If you calculated $d = a \pmod N$ where $N$ is another very large number is it possible to prove that $ c_1 \mid a$ without knowing what $a$ is? All you know in this situation is $d, N$ and $c_1$.
I hope this question makes sense, please let me know if it doesn't.