Prove the following theorems Theorem For x Z if x 3 1 is odd

Prove the following theorems.

Theorem For x Z, if x 3 1 is odd, then x is even.

Theorem If x 2 is a prime number, then x is not an integer.

A) We will prove it by contradiction.

Suppose x is odd and X^3 -1 is odd.

=> X^3 = odd+1

=> X^3 is even

But cube of a odd number cannot be even. So x must be even.

b) We will also prove it by contradiction.

Lets assume x is integer and x^2 is a prime number.

But if x is integer then x*x will always be non prime as it will have x as divisor.

So x must not be an integer.