Suppose that p is a prime number, a and b are integers, and p divides both ab and a + b. Prove that p divides both a and b

This can be used for proving above:
For all primes p and all integers a, b, if p | ab, then p | a or p | b (or both)

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Given that p divides ab then either a or b or both are multiple of p.
(a) Case 1 : a is multiple of p i.e. there is some integer m such that a = pm
and p divides a + b then...
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