Each of X and Y is a positive integer with X ≤ Y. The quotient and the remainder obtained upon dividing X² + Y² by X+Y are respectively denoted by Q and R.

Determine all possible pairs (X, Y) such that Q² + R = 1977

Supplementary questions:

This problem has been out of circulation for quite some time. Why? When is it likely to come back into favour?

I have the set (7,50).
This yields:
 X² +Y²  =  2549   and X + Y = 57

The quotient Q²  (1936) + R (41) = 1977.

Variances within X and Y cause shifts around that value,
eg  X=  9 and Y = 51  yields  1978
     X =15 and Y = 53  yields  1978.
                                                   It would therefore seem that no other values fit that scenario.


For the Supplementary question I'm assuming that we are to look for the next year AD where a unique pairing only will yield that year.

Posted by brianjn on 2011-02-08 22:07:54