Why does an even number plus an odd number always equal an odd number?

Answer 1

For any even number n, the following integer is odd (n+1), so for any odd number other than 1, you are adding the preceding even number, plus 1. Because all even numbers are multiples of 2, their sum is even, and 1 more will be an odd number.

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Think of the odd number as an even number plus one and remember that all even numbers are multiples of two.

Therefore:- Any two even numbers added together will always be a multiple of two and therefore even. Any two odd numbers added together will be the same as adding two even numbers plus 1 plus 1. (A multiple of two plus two which is still a multiple of two).

An odd number added to an even number will be the same as two even numbers plus 1 in other words a multiple of two (even) plus one making the result odd.

Extrapolating further it's easy to see that by the above reasoning that for all positive integers...

1. Any number of even numbers added together will always be even.
2. An odd number of odd numbers added together will always be odd.
3. Any even number of odd numbers added together will always be even,
4. Adding any number of even numbers to any number of odd numbers will make no difference to the outcome, the odd or even outcome will still be determined by the number of odd numbers being added together.

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Another approach:

Let n, r be real numbers even number representation: 2n

odd number representation: 2n + 1

sum = 2n + 2n + 1 = 4n + 1 = 2(2n) + 1

let 2n = r

sum representation becomes: 2r + 1, which is a representation of an odd number.