Find all possible positive integer solutions to this system of simultaneous equations:

AB+CD = 2016, and:
AD - BC = 1

*** For an extra challenge, solve this puzzle without a computer program aided method.

Using the identity  (A² + C²)(B² + D²) = (AB + CD)² + (AD - BC)²,

       (A² + C²)(B² + D²)   = 2016² + 1²

                                    = 4064257

                                    = 317 x 12821  (both primes)     *

                                    = (11² + 14²)(70² + 89²)           *

Since 14 x 70 > 11 x 89, of the four possible allocations of these
numbers to A, B, C, D, only two satisfy the original equations: viz

{A, B, C, D} = {14, 89, 11, 70}  or  {A, B, C, D} = {70, 11, 89, 14}

Done without a computer? Well not quite (*), but Srinivasa would
have no trouble.

Edited on May 26, 2016, 7:44 pm

Edited on May 26, 2016, 7:45 pm

Posted by Harry on 2016-05-26 19:44:07