If x and y are positive integers, and are not both even which of the following must be even?

Is it
4(x + y) -2

4(x + y) -2

= 2( 2(x+y) - 1)

any integer multiplied by 2 is clearly even, so .....

It is correct?

I think I just said so.

To determine whether 4(x + y) - 2 is even or odd, we need to see if the result is divisible by 2.

Let's consider two cases:

Case 1: Both x and y are odd numbers.
If both x and y are odd numbers, then the sum (x + y) will be an even number. This is because the sum of two odd numbers is always even. Therefore, 4(x + y) will also be even since multiplying an even number by any other number gives an even result. Finally, subtracting 2 from an even number will also result in an even number. Therefore, when x and y are odd numbers, 4(x + y) - 2 is even.

Case 2: One of x or y is odd and the other is even.
If one of x or y is odd and the other is even, then the sum (x + y) will be odd. This is because the sum of an odd number and an even number is always odd. Therefore, 4(x + y) will be even since multiplying an even number by any other number gives an even result. Lastly, subtracting 2 from an even number will still result in an even number. Therefore, when one of x or y is odd and the other is even, 4(x + y) - 2 is even.

In both cases, we have shown that 4(x + y) - 2 is even. So, regardless of whether x and y are odd or one is odd and the other even, the expression 4(x + y) - 2 will always be even.