How many ordered pairs (x, y) of positive integers satisfy 1/x + 1/y = 1/12?

Clear denominators: 12y + 12x = xy → xy − 12x − 12y = 0 → (x − 12)(y − 12) = 144.

Since 1/x < 1/12, x > 12, and similarly y > 12, so both factors are positive.

So the number of ordered pairs = number of positive divisors of 144 = 2⁴ × 3² → (4 + 1)(2 + 1) = 15.

Tip: For 1/x + 1/y = 1/n, the number of ordered positive solutions equals the number of divisors of n².