Numerical answer
How many ordered pairs of integers (x, y) satisfy x² − y² = 96?
Show answer and solution
Answer: 16
Write (x − y)(x + y) = 96. Let a = x − y and b = x + y; then x = (a + b)/2 and y = (b − a)/2, so a and b must have the same parity.
Since 96 is even, both must be even: a = 2m, b = 2n with mn = 24.
24 has 8 positive divisors, so there are 8 ordered positive pairs (m, n), and another 8 with both negative, giving 16 ordered pairs (a, b). Each yields a distinct integer pair (x, y).
Answer: 16.
Trap: Counting all divisor pairs of 96 (24 ordered with signs) overcounts, since odd × even pairs such as (1, 96) or (3, 32) give non-integer x, y.
Since 96 is even, both must be even: a = 2m, b = 2n with mn = 24.
24 has 8 positive divisors, so there are 8 ordered positive pairs (m, n), and another 8 with both negative, giving 16 ordered pairs (a, b). Each yields a distinct integer pair (x, y).
Answer: 16.
Trap: Counting all divisor pairs of 96 (24 ordered with signs) overcounts, since odd × even pairs such as (1, 96) or (3, 32) give non-integer x, y.
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