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How many ordered pairs (a, b) of positive integers satisfy 1/a + 1/b…
Question
How many ordered pairs (a, b) of positive integers satisfy 1/a + 1/b = 1/6?
✓ Verified answer: 9checked by our engine — not a guess
Step-by-step solution
From 1/a + 1/b = 1/6 we get 6(a+b) = ab, i.e. ab - 6a - 6b = 0. Add 36 to both sides: (a-6)(b-6) = 36.
We need a, b positive integers; since a > 6 and b > 6 are forced (each factor must be a positive divisor of 36), the ordered factorizations of 36 = (a-6)(b-6) correspond exactly to ordered pairs of positive divisors.
The number of positive divisors of 36 = 2^2·3^2 is (2+1)(2+1) = 9.
Each divisor d of 36 gives a-6 = d, b-6 = 36/d, a valid positive-integer pair. Hence there are 9 ordered pairs.
Final answer9
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