Proving if something is even

Let A,B,C be integers such that A^2 + B^2 = C^2. Prove at least one of A and B is even.

Please help guide me through this problem so I can understand it better thanks.

I have another question also

let a and b be positive such that a^2 = b^3

given 4 divides b, prove that 8 divides a

Im hoping if someone can guide me through this problem also thanks.

Re: Proving if something is even

For the first question, have you taken a look at Euclid's method for generating Pythagorean triples?

For the second question, I would let b = 4n...what do you find?

Re: Proving if something is even

for the first one i want to try to use a proof by contradiction but i am kind of stuck on how to do it

and for the 2nd one when b = 4n

b^3 = (4n)^3

so a^2 = (4n)^3

so a = sqrt((4n)^3)

Re: Proving if something is even

What is the square root of 4 cubed?

Re: Proving if something is even

Edit: Nevermind I figred everything out thanks a lot for your help!

Re: Proving if something is even

Hello, gfbrd!

First, we will establish a theorem.

Consider the square of an integer,

If is even, we have: .

. . Hence: .

If is odd, we have: .

. . Hence: .

Therefore, the square of an integer is either:

. . (1) a multiple of 4, or

. . (2) *one more* than a multiple of 4.

We are given: .

Suppose and are *both** *__odd__.

We have: .

The we have:

. .

. . . . . . . . .

. . . . . . . . .

Hence: is *two* more than a multiple of 4.

. . . . . .It can*not* equal a square,

Therefore, at least one of and must be even.