Show that if n divides m where n and m are positive ints greater than 1, then a ≡ b (mod m) implies a ≡ b (mod n) for any positive integers a and b

Question

Show that if n divides m where n and m are positive integers greater than 1, then a ≡ b (mod m) implies a ≡ b (mod n) for any positive integers a and b.

I recognise that a similar question has been posted here a year ago, but I really don't understand the answers provided. This is part of my discrete math's assessment, so I kind of need to show my work, and to do so, I got to understand what am doing. Any help would be greatly appreciated.

Answer 1

$$a \equiv b \pmod m \iff m \mid a-b \implies n \mid a-b \iff a \equiv b \pmod n$$