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2016 and Simultaneous Settlement (Posted on 2016-05-26) Difficulty: 3 of 5
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.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Analytic Solution Comment 2 of 2 |
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

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