- #1
Adel Makram
- 635
- 15
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.