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This problem trains for: SAT-I, AMC-8, GMAT.

The difference of the squares of two positive integers is 72. What is the largest possible sum of the two positive integers?

Denote the two integers by p and q. The condition can be written as:

equation

We see that this is, in fact, a Diophantine equation with two integer unknowns. It can be easily solved using the fundamental theorem of arithmetic (the prime factorization of an integer is unique). Factoring both sides:

equation

We are looking for the largest value of p+q.

Since the sum and the difference of two integers have the same parity (are either both odd or both even), we must have at least one factor of 2 in each.

Therefore, the smallest possible difference is 2 and the largest possible sum is 36.