Numerical answer
How many ordered pairs of positive integers (x, y) satisfy 1/x + 1/y = 1/12?
Show answer and solution
Answer: 15
Clear denominators: 12y + 12x = xy → xy − 12x − 12y = 0 → (x − 12)(y − 12) = 144.
Both x and y must exceed 12 (each of 1/x, 1/y must be less than 1/12), so x − 12 and y − 12 are positive divisors of 144 whose product is 144.
144 = 2⁴ × 3² has (4 + 1)(2 + 1) = 15 divisors, each giving one ordered pair.
Answer: 15.
General rule: 1/x + 1/y = 1/n has d(n²) ordered positive solutions.
Both x and y must exceed 12 (each of 1/x, 1/y must be less than 1/12), so x − 12 and y − 12 are positive divisors of 144 whose product is 144.
144 = 2⁴ × 3² has (4 + 1)(2 + 1) = 15 divisors, each giving one ordered pair.
Answer: 15.
General rule: 1/x + 1/y = 1/n has d(n²) ordered positive solutions.
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