Square root pair
Algebra
Level
3
Find the number of positive integer pairs such that
The answer is 4.
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1 solution
a + 2 b = a − 2 b + b a + 2 b − a − 2 b = b ( a + 2 b − a − 2 b ) 2 = b 2 a − 2 a 2 − 4 b = b 2 a − b = 2 a 2 − 4 b ( 2 a − b ) 2 = 4 ( a 2 − 4 b ) 4 a 2 − 4 a b + b 2 = 4 a 2 − 1 6 b b 2 − 4 a b + 1 6 b = 0 b ( b + 1 6 − 4 a ) = 0 b = 4 a − 1 6 = 4 ( a − 4 ) ∴ 4 ∣ b 2 a − b = 2 a 2 − 4 b ≥ 0 2 a − b = 2 × 4 b + 1 6 − b = 2 b + 8 − b ≥ 0 b ≤ 1 6 ∴ b = { 4 , 8 , 1 2 , 1 6 }