DashJ

Why is it that when you add two even numbers together, the result remains even, but two odd numbers also become even?

• EVEN NUMBERS can be looked at as any number (call it "n"), multiplied by 2. Therefore, all even numbers can be described as 2n. Two even numbers added together can be written as: 2n + 2m, where n and m are the even numbers in question, divided by two. A simple rearranging of the terms above gives: 2n + 2m = 2(n + m). Therefore, any even number plus any other even number will always equal an even number (as the answer you get will always be some number multiplied by two). An odd number can be looked at as an even number with one added to it - e.g. 5 is 4+1. Therefore, if you add two odd numbers together, what you're really doing is adding an even number to another even number, then adding 1 + 1, which is 2, and therefore even. As shown above, adding three even numbers together will always give an even number. QED.

  Simon Cooke, (scooke@nessie.mcc.ac.uk)

• Here's maybe a slightly less algebraic-sounding explanation: An odd number by definition leaves a remainder of one when it's divided by two. If I add this odd number to another odd number, I'm left with two remainders of 1, which of course add to two, thus cancelling out those remainders. So the sum is even. An even number by definition has no remainder when divided by two. So adding it to another even number will still generate no remainder. Hence an even result.

  Jay Weedon, New York, USA