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Dozen Divisibility Dilemma (Posted on 2010-12-02) Difficulty: 3 of 5
Each of x and y is a positive integer such that: x2 5y2 = 1.

Prove that x*y is always divisible by 12.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts re: Possible solution | Comment 3 of 4 |
(In reply to Possible solution by broll)

I agree with Jer; your 'elementary' method is interesting and worth perfecting, but the 'oddness' of x still needs to be proven. Would this do the trick?...

Assuming x is even, write x = 2X, so that x2 - 5y2 = 1 becomes 4X2 - 5y2 = 1.

Then working modulo 4, this gives        y2 = 3 (mod 4).

Now, squares can only take the values 0 or 1 (mod 4), so we have a contradiction, proving that if a solution exists for this equation, then x is odd.

Edited on December 3, 2010, 9:28 pm
  Posted by Harry on 2010-12-03 21:26:03

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