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## Homework Statement

Prove that if a right triangle has all sides rational and primitives (co-primes), then one of the smaller side must be even number.

## Homework Equations

For a right triangle (a,b,c) with c is the hypotenuse.

$$a^2+b^2=c^2$$

## The Attempt at a Solution

In order to create a contradiction, I assume both a and b are odd, so.

$$a=2n_1 +1$$

and.

$$b=2n_2+1$$

applying Pythagorean theorem,

$$a^2+b^2=4(n_1^2+n_2^2-n_1-n_2)+2$$.

This only gives me that c must be even but it does not tell me whether it is still rational and co-primes to a and b or not.