Pythagorean triples

**zhupolongjoe** · January 27th 2010, 11:34 AM

Show that the area A of a triangle whose sides form a pythagorean triple is always divisible by 6.

Hint: For an integer x, x^2 is of the form 4k or 1+8k, and when 3 doesn't divide x, x^2 is of the form 1+3k.

I don't know what to do. I figured the strategy would be to show that it is divisible by 2 and also by 3, so that it is divisible by 6=2*3, but I don't know how to go about this or if it is a correct approach.

thanks

**Dinkydoe** · January 27th 2010, 12:51 PM

I hope you can do a little modular arithmatic. You're idea is indeed the right one:

Let $(a,b,c)$ be such that $a^2+b^2= c^2$
The area of the triangle is given by: $A = \frac{1}{2}ab$

Observe that: $c^2\equiv 0,1$ mod 4. And $c^2\equiv 0,1$ mod 3. (you can easily check this by looking at quadratic residu-classes modulo 4, and modulo 3).

Case 1.
Assume $c^2=a^2+b^2\equiv 0$ mod $4 \Longrightarrow a^2\equiv b^2\equiv 0$ mod 4 $\Longrightarrow ab\equiv 0$ mod 4.

Assume $c^2= a^2+b^2\equiv 0$ mod 3 $\Longrightarrow a^2\equiv b^2\equiv 0$ mod 3 $\Rightarrow ab\equiv 0$ mod 3.

Case 2.
Assume $c^2 = a^2+b^2\equiv 1$ mod 4. This implies $a^2+b^2\equiv 1$ mod 8, or $a^2+b^2\equiv 5$ mod 8

Now we'll look at quadratic residues modulo 8 wich are $\left\{0,1,4\right\}$

Since there is no quadratic residu that is 5 mod 8. It follows that $c^2=a^2+b^2\equiv 1$ mod 8 $\Rightarrow a^2\equiv 0$ mod 8, $b^2\equiv 1$ mod 8. Hence $ab\equiv 0$ mod 8.

Assume $c^2= a^2+b^2\equiv 1$ mod 3 $\Longrightarrow a^2\equiv 0$ mod 3, $b^2\equiv 1$ mod 3 $\Rightarrow ab\equiv 0$ mod 3

So in both cases we have $4|ab$ and $3|ab\Rightarrow 6|A$

**zhupolongjoe** · January 27th 2010, 01:54 PM

Thanks so much!

**Dinkydoe** · January 27th 2010, 02:54 PM

Please check my corrections:

I argued that if $c^2 = a^2+b^2\equiv 1$ mod 4 that the only possiblility is that $c^2=a^2+b^2\equiv 1$ mod 8

This is an important observation!

**zhupolongjoe** · January 29th 2010, 02:18 PM

I'm sorry, I have one more question. I understand your proof, but wouldn't it be easier to do it with mod 3 and mod 2 instead of mod 3 and mod 4, or is there some reason why you can't consider it with mod 2?

Thanks.

**Dinkydoe** · January 29th 2010, 03:25 PM

We want to proof that $2|(\frac{ab}{2})$ hence we must prove $4|ab$ . That's why $ab\equiv 0$ mod 2 isn't enough.

**zhupolongjoe** · January 29th 2010, 03:31 PM

Sorry for all of the questions, but then wouldn't we need to show that 6 divides ab instead of 3 since we need to show that 3 divides (ab)/2.

**Dinkydoe** · January 30th 2010, 03:43 AM

Sorry for all of the questions, but then wouldn't we need to show that 6 divides ab instead of 3 since we need to show that 3 divides (ab)/2

No. if $3|ab$ then it follows $3|\frac{ab}{2}$ , since 2,3 are coprime. Do I need to proof this for you? ;p

If we show that $3|ab$ and $4|ab$ then we've shown that $12|ab$ . Hence $6|{\frac{ab}{2}}$ .

**zhupolongjoe** · January 30th 2010, 06:21 AM

Ok, I've got it now. Thanks for your patience.

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