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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

 See The Solution Submitted by K Sengupta Rating: 2.0000 (2 votes)

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 My start | Comment 2 of 4 |

We want a and b such that (a+b)*q + r = a^2 + b^2

With q^2 + r = 47 the possibility that q=50 and r=47 is hard not to notice so I tried it.

The resulting equation can be written as

a^2 - 50a + (b - 50b - 47) = 0 which is a quadratic in a.

There are three solutions for (a,b) using brute force:

(61,24) and (61,26) are the two that works.  (23,2) does not.

This is obviously not the right approach as it uses brute force and only uses one possibility for q and r.  But I did find some solutions at least.

 Posted by Jer on 2007-03-30 11:30:41

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