Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$ Would this be the same thing as saying "Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for any $s, t \in \mathbb{Z}$"? I can do the proof for any integers $s$ and $t$, but if any and all mean the same thing, then I can do this proof. If not, how can I do this?

Assume that $a|b$ and $a|c$, i.e., $\frac ba$ and $\frac ca$ are integers. For any $s,t\in\mathbb Z$ we have
$$\frac{sb+tc}a=s\left(\frac ba\right)+t\left(\frac ca\right)\in\mathbb Z$$
(since sums and products of integers are integers), so $a|(sb+tc)$.
