2007 AMC 10A Problem 23

Question

How many ordered pairs

(m, n)

of positive integers, with

m \gt n,

have the property that their squares differ by

96?

3

4

6

9

12

Write

m^2 - n^2 = (m + n)(m - n) = 96

m + n

and

m - n

have the same parity, so both must be even

The factors

m + n

and

m - n

have the same parity. Since their product is

96,

they cannot both be odd, so both must be even.

The even factor pairs are

(48, 2),

(24, 4),

(16, 6),

and

(12, 8),

giving

(m, n) = (25, 23),

(14, 10),

(11, 5),

and

(10, 2).

So there are

4

ordered pairs.

Thus, the correct answer is B.

Problem 23 in other years