q) is true, we assume that p is true and show that q is also
true.5. To prove that an if--then statement (p q) is true, we assume that p is true and show that q is also
true.
Proof Suppose that k, a and b are integers such that k | a and k | b.
| By the definition of divisibility: | If k | a, | then a = k · s | for some integer s. |
| If k | b, | then b = k · t | for some integer t. |
| Adding the equations for a and b we see that: | a + b |
= |
k·s + k·t |
= |
k (s + t) |
Since s and t are both integers, we know that s + t is also an integer. Therefore a + b = k·r for some integer r. Hence k | (a + b).