Theory of numbers – Verification of divisibility in different modules.

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

Assume 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 to check if $ d mod n = 0 $ It does not work, but it is impossible to perform this check or there is a general way to find out if $ c | a $? Maybe yes $ m $ Y $ n $ they are cousins?

Thank you!