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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

Prove that if

then 60|xyz (where x, y and z are positive integers).Offline

**JaneFairfax****Member**- Registered: 2007-02-23
- Posts: 6,868

Pythagorean triples are of the form

where

are positive integers with .So we want to show that

is divisible by 30, i.e. by 2, 3 and 5.

**2***N* is even if one of *a* and *b* is even. If theyre both odd, then their sum is even, so *N* is again even.

**3**

If neither *a* nor *b* is divisible by 3, then one of the following happens:

In cases (i) and (ii),

; in case (iii), .**5**

Suppose

Then by Fermats little theorem,

; .QED.

*Last edited by JaneFairfax (2008-11-27 14:56:23)*

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**Daniel123****Member**- Registered: 2007-05-23
- Posts: 663

Nice I did it slightly differently. I considered the possible values of x, y and z mod 3, mod 4 and mod 5.

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