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2007 AMC 10A Problem 23

Problem 23 of 25HarderAlgebraNumber Theory

How many ordered pairs (m,n)(m, n) of positive integers, with m>n,m \gt n, have the property that their squares differ by 96?96?

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Solution

The factors m+nm + n and m−nm - n have the same parity. Since their product is 96,96, they cannot both be odd, so both must be even. The even factor pairs are (48,2),(48, 2), (24,4),(24, 4), (16,6),(16, 6), and (12,8),(12, 8), giving (m,n)=(25,23),(m, n) = (25, 23), (14,10),(14, 10), (11,5),(11, 5), and (10,2).(10, 2). So there are 44 ordered pairs. Thus, the correct answer is B.

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Concepts: difference of squares · Diophantine Equation · parity

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.