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How many ordered pairs {a, b} with integers a and b satisfy 2a² + 3b² = 35, where Z denotes the set of all integers?

1. 2
2. 4
3. 8
4. 12

Correct answer: 8

Solution

Testing b^2 in {0,1,4,9}: 3b^2 = 3 leaves 2a^2 = 32 -> a = +/-4, and 3b^2 = 27 leaves 2a^2 = 8 -> a = +/-2. So (a,b) = (+/-4,+/-1) and (+/-2,+/-3), giving 8 ordered integer pairs.

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