Math please help!?

Question

how we show If a | b and a | c, then a | (b + c)
a and If a | b or a | c, then a | (b * c)

Answer 1

Suppose a|b and a| c. Then, there exists integers k1 and k2 such that b = k1 * a and c = k2 * a, which implies b + c = k1 * a + k2 * a = (k1 + k2)a. Since k1 and k2 are integers, so is k1 + k2. Therefore, a|(b + c).

Now, suppose a|b (if a|c, the proof is exactly the same). Then, there exists an integer k such that b = k *a. It follows b*c = k *a *c = a*(k*c). Since k and c are integers, so is k*c. Therefore, a|(b*c).

Answer 2

If a|b, then ak = b
If a|c then ah = c


adding both sides gives you ak+ah = b+c
which factors to a(k + h) = b+c
therefore a|b+c

You can write a similar proof for your second statement.