proof of modular congruencies

Question

Is it possible to show that given $a,b,d,n \in \mathbb{N}$, if $a \equiv b \pmod{m}$ and $d\mid n$, then $a \equiv b \pmod{d}$?

I'm able to see it computationally by checking numbers but can't seem to formulate a proof so I'm not sure if I'm just picking lucky numbers that work out?

I'm trying to use this as a intermediate step in a larger proof but am unsure if it's valid.

score 2 · Answer 1 · answered Sep 14 '16 at 23:17

2

A divisor of a divisor is a divisor, so $$a\equiv b\mod n\iff n\mid a-b\implies d\mid a-b\iff a\equiv b\mod d.$$

answered Sep 14 '16 at 23:17

Bernard

175,478

oh wow that was far more trivial than i was making it. thanks. – greenteam Sep 14 '16 at 23:19
Beginners often tend make things more complex than what they really are. In my opinion, the first line would be enough. – Bernard Sep 14 '16 at 23:21

proof of modular congruencies

1 Answers1