Numerical answer
How many ordered pairs of positive integers (x, y) satisfy 1/x + 1/y = 1/6?
Show answer and solution
Answer: 9
Clear the fractions: 6y + 6x = xy → xy − 6x − 6y = 0 → (x − 6)(y − 6) = 36.
Since 1/x < 1/6, x > 6, and likewise y > 6. So both factors are positive.
The number of ordered pairs equals the number of positive divisors of 36 = 2² × 3², which is (2 + 1)(2 + 1) = 9.
For example: (7, 42), (8, 24), (9, 18), (10, 15), (12, 12), and the reverses of the first four.
Tip: 1/x + 1/y = 1/n always has d(n²) ordered positive solutions.
Since 1/x < 1/6, x > 6, and likewise y > 6. So both factors are positive.
The number of ordered pairs equals the number of positive divisors of 36 = 2² × 3², which is (2 + 1)(2 + 1) = 9.
For example: (7, 42), (8, 24), (9, 18), (10, 15), (12, 12), and the reverses of the first four.
Tip: 1/x + 1/y = 1/n always has d(n²) ordered positive solutions.
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