Math Is Fun Forum

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).

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.

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

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Number theory proof

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