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#1 2008-11-27 06:56:55

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

Number theory proof

Prove that if

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

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#2 2008-11-27 14:40:28

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

Re: Number theory proof

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 they’re 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 Fermat’s little theorem,

;
.

QED. smile

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


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#3 2008-11-28 02:22:53

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

Re: Number theory proof

Nice smile 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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