Question

Prove: If a^2-2a is even then a is even

Answer 1

Prove by contradiction

Explanation:

If
`a^2-2a` is even then
`a` is even

Prove by contradiction

`a^2-2a` is even then
`a` is odd

Also
`a^2-2a=a(a-2)`

Since
`a` is odd
`a-2` is also an odd number, the addition or subtraction of an even number with an odd number is an odd number.

The multiplication of two odd numbers yields to another odd number, then
`a(a-2)` is odd but is equal to
`a^2-2a` which is even, a number cannot be odd and even, this is a contradiction so
`a` must be even.