How many ordered pairs of positive integers (x, y) satisfy x² − y² = 175?

Factor: (x − y)(x + y) = 175. Since x, y are positive integers with x > y, both (x − y) and (x + y) are positive integers, and (x + y) > (x − y).

175 = 5² × 7, so its divisors are 1, 5, 7, 25, 35, 175 — all odd (as 175 is odd), so every factor pair has matching parity and gives an integer solution.

Pair up d = x − y with 175/d = x + y, taking d < 175/d: (1, 175), (5, 35), (7, 25).

Each gives x = (d + 175/d)/2, y = (175/d − d)/2:
- (1,175) → (x,y) = (88, 87)
- (5,35) → (x,y) = (20, 15)
- (7,25) → (x,y) = (16, 9)

All three check out (88² − 87² = 175, 20² − 15² = 175, 16² − 9² = 175), giving 3 ordered pairs.

Tip: For x² − y² = N with N odd, the count of positive-integer solutions equals the number of divisors of N less than √N (each paired with its cofactor).