If a,b,c,d are integers and p is a prime factor of both a-b and c-d, prove that

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By contradiction, let d>1 divide both ab and a+b. Then d has a prime divisor p, which divides both ab and a+b. Then p divides a(a+b) - ab = a^2 Since p is a prime which divides a*a, we have that p divides a. And p divides b(a+b) - ab = b^2 Similarly, p divides b. Hence p divides a and p divides b, so p divides gcd(a,b) = 1. However, this forces p = 1, which is a contradiction, since 1 isn't a prime. Thus the only positive common divisor of ab and a+b is 1, which shows they are coprime.

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