I want to prove the following statements:
An odd positive integer raised to a positive power results in an odd number.
An even positive integer raised to a positive power results in an even number.
However, I am not sure how to do this mathematically. Specifically, for the first statement, I can explain it’s true because no matter how many times you multiply odd numbers, you will continue getting odd numbers, but this doesn’t seem very rigorous.
For the second statement, my proof is more rigorous as I say that
(2m)$^n$ = 2$^n$m$^n$, which must be even as 2$^n$ is always even.
Any comments regarding whether these proofs are valid would be great.