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The two positive integers a, b with a> b are such that a% of b% of a and b% of a% of b differ by 0.003. Find all possible pairs (a, b).
There are four possible pairs: (5, 2), (5, 3), (6, 1) and (6, 5).
Since a and b are integers, we know a, b and a− b are all factors of 30.
Further, we know that a < 10, since if a ≥ 10 then one of b and a - b ≥ 5, which would make theproduct too big.
We also know that a (being the largest of the threenumbers a, b and a - b) must be greater than 301/3 > 271/3= 3.
Hence a, being a factor of 30 between 3 and 10, can only be 5 or 6.
If a = 5, then
b (5 − b) = 6
b2 − 5b + 6 = 0
(b − 2)(b − 3) = 0
b = 2 or b = 3
If instead a= 6, then
b (6 − b) = 5
b2 − 6b + 5 = 0
(b − 1)(b − 5) = 0
b = 1 or b = 5
Hence, in total, there are four possible pairs of (a, b): (5, 2), (5, 3), (6, 1) and (6, 5).
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