# 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!