An even number can be divided by 2 evenly. An odd number will have a remainder of 1 when divided by 2. Even plus odd is odd.
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The sum of two odd numbers is always even.
The sum of two odd numbers is always even.
Odd + even = odd
Because the sum of two odd numbers is always even and the sum of that even number plus another odd number is always odd.
Count how many of the numbers are odd. If an odd number of them are odd, then their sum is odd. If an even number of them are odd, then their sum is even.