number theory – Verification of divisibility in different modules

So I'm working on modular arithmetic and I have encountered the following problem:

Suppose that $ a cdot b mod m = c $ and that I know $ c mod n = d $. Can I still check the divisibility of $ c $ using $ d $? A quick check tells me that check if $ d mod n = 0 $ does not work – but is this verification impossible or is there a general way of knowing if $ c | a $? Can be so $ m $ and $ n $ are prime numbers?

Thank you!