Math Is Fun Forum
  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

Login

Username

Password

Not registered yet?

#1 2008-11-28 05:56:55

Daniel123
Power Member

Offline

Number theory proof

Prove that if

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

 

#2 2008-11-28 13:40:28

JaneFairfax
Legendary Member

Offline

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 (2008-11-28 13:56:23)


Q: Who wrote the novels Mrs Dalloway and To the Lighthouse?

A: Click here for answer.
 

#3 2008-11-29 01:22:53

Daniel123
Power Member

Offline

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.

 

Board footer

Powered by FluxBB