By definition, $a|b$ means $b = ak$ and $b|c$ means $c = bm$. Substituting, $c = (ak)m = a(km)$, so $a|c$ by definition.

Any two consecutive multiples are divisible by each other.

Divisibility is an equivalence relation, so it is transitive by default.

The remainder when dividing $c$ by $a$ must equal zero.