If $a \equiv b \pmod n$ and if $d \mid n$, then $a \equiv b \pmod d$
It is simple as just need the divisibility based relation, $n = dk, \exists k \in \mathbb{Z}$.
As, $a - ln = b - mn, \exists l,m \in \mathbb{Z}$, so for any factor $d$ of $n$ it should also hold true, i.e. :
$a - lkd = b - mkd, \exists k,l,m \in \mathbb{Z}$.
My issue is should the difference be in terms of $lkd, mkd$ only, or something else is also possible. Any small working example will show that only $lkd, mkd$ are the correct choices.