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 A 2547 Puzzle (Posted on 2007-03-29)
a and b are positive integers. Dividing a2 + b2 by a + b we obtain the quotient as q and the remainder as r.

Determine analytically all possible pairs (a, b) such that q2 + r = 2547

 Submitted by K Sengupta Rating: 2.0000 (2 votes) Solution: (Hide) a^2 + b^2 >= (a + b)^2/2, so q >= (a + b)/2. Hence r < 2q. The largest square less than 2547 is 2500 = 50^2 and: 2547 = 50^2 + 47. The next largest gives 2547 = 49^2 + 146 . But 146 > 2*43. So we must have q = 50, r = 47. Hence a^2 + b^2 = 50(a + b) + 47 So, (a - 25)^2 + (b - 25)^2 = 1297 = 36^2 + 1^2 Accordingly, (a-25, b-25) = (36, 1); (36, -1); (1, 36); (-1, 36), since both a and b are positive integers. Consequently, (a,b) = (61, 26); (61, 24); (26,61); (24, 61)

 Subject Author Date positive- yes integers-no Ady TZIDON 2007-03-31 11:34:06 re: My start Ady TZIDON 2007-03-31 10:43:18 My start Jer 2007-03-30 11:30:41 a start Dej Mar 2007-03-30 00:54:54

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