The words "even" and "odd" have several meanings in mathematics. If we're talking about numbers, an even number is an integer that's divisible by 2 with nothing left over. Yes, it's a major bummer when there are no leftovers, but even numbers aren't really that delicious in the first place, so it's not a huge loss.
Here are some even numbers:
...-6, -4, -2, 0, 2, 4, 6...
Yep, 0 is even, because 0 can be divided by anything with nothing left over. That means 0 ÷ 2 = 0, so it's even.
If we were to continue filling in that list in both directions forever, we would get all the even numbers. Although, since there's an infinite amount of them, we'd need an awfully long scroll bar over there →
An odd number is any integer that's not divisible by 2. Try to divide these guys by 2 and we'll always have 1 or -1 left over. (Great; that'll be tomorrow's lunch.) In other words, any integer that's not even is odd. Here are some examples:
...-7, -5, -3, -1, 1, 3, 5, 7...
Every integer must be either even or odd, 'cause every integer is either divisible by 2 or not divisible by 2. There are no other possibilities. And that's not just us being pessimistic. It's a fact, yo.