Prove that if $a|b$ and $a|c$ then $a|(sb+tc)$ for all $s, t \in \mathbb{Z}$

In response to your question about any and all, they generally mean the same thing. The symbol $\forall$ can be read "for all" or "for every" or "for any" or "for each". These all mean the same thing.

This is how you could write the proof, with meticulous rigor.

Let $a,b,c \in \mathbb{Z}$ with $a \mid b$ and $a \mid c$.

Let $s,t \in \mathbb{Z}$ be arbitrary.

Then by definition, $\exists\, k_1, k_2 \in \mathbb{Z}$ such that $$b = ak_1$$

$$c = ak_2$$

Then $$sb = s(ak_1)$$

$$tc = t(ak_2)$$

By commutativity and associativity,

$$sb = a(sk_1)$$

$$tc = a(tk_2)$$

Adding these equations, we get

$$sb + tc = a(sk_1) + a(tk_2)$$

By distributivity,

$$sb + tc = a(sk_1 + tk_2)$$

Since $\mathbb{Z}$ is closed under addition and multiplication, $$(sk_1 + tk_2) \in \mathbb{Z}$$

Thus, by definition, $$a \mid (sb+tc)$$

try writing both numbers as a combination of integers and remainders b=qa+r c=q'a+r sb+tc=sqa+sr+tq'a+tr=a(sq+sq')+r(s+t) since a divides b, c, and 0 <= r <=a , we conclude r must be 0. from here, forall s,t in Z, a divides any linear combination of b and c

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 $a|(sb+tc)$.