a and b are positive integers. Dividing a^{2} + b^{2} by a + b we obtain the quotient as q and the remainder as r.
Determine analytically all possible pairs (a, b) such that q^{2} + r = 2547
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 20070330 11:30:41 