Prove that if the integer \(\displaystyle k\) divides the integers \(\displaystyle a\) and \(\displaystyle b\) then \(\displaystyle k\) divides \(\displaystyle as + bt\) for every pair of integers \(\displaystyle s\) and \(\displaystyle t\).
If we assume that the integer \(\displaystyle k\) divides the integers \(\displaystyle a\) and \(\displaystyle b\), then we have:
\(\displaystyle m = \frac{a}{k}\) and \(\displaystyle n = \frac{b}{k}\)
Or
\(\displaystyle a = mk\)
\(\displaystyle b = nk\)
where \(\displaystyle m,n \in \mathbb{Z}\).
Let \(\displaystyle s,t\) be arbitrary integers, then
\(\displaystyle as + bt = mks + nkt = k(ms + nt)\)
Since \(\displaystyle m, n, s, t \in \mathbb{Z}\) and sums and products of integers are integers, then \(\displaystyle (ms + nt) \in \mathbb{Z}\)
Since \(\displaystyle as + bt = k \times \text{integer}\), then \(\displaystyle k\) divides \(\displaystyle as + bt\) for every pair of integers \(\displaystyle s\) and \(\displaystyle t\).
