Let x and y be integers Show that x and y are both even iff

Let x and y be integers. Show that x and y are both even iff xy and x +y are both even.

Solution

Given:

xy is even and x+y is even

For xy to be even at least one of x and y must be even as odd*odd = odd but even*even and even*odd will give us even. So, at least one of them is even.

Also, x+y is even. Sum of two odd or two even numbers is even because sum of odd and even will be odd always. So, either both are odd or both are even. But we proved above that at least one is even. So, the statement both even must hold true for both the conditions to hold true.