Why do two even numbers always equal an even number?

Answer 1

On a number line, adding an even number to another number (or zero) results in an even displacement, which must end in the same type of number as the original.

If the beginning number is odd, adding an even number produces an odd sum.

If the beginning number is even, adding an even number produces an even number.

A corollary is that:

Adding two like numbers produces an even number.

Adding two unlike numbers produces an odd number.

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Algebraically:

let x be an even number,

Then y = 2x for some value of x

Two even numbers would be 2m and 2n

Adding them gives:

2m + 2n = 2(m + n) = 2p where p = m + n;

2p is of the form y = 2x, so 2p is an even number.

Thus adding two even numbers results in an even number.

Similarly for odd numbers:

If y = 2x is an even number then z = y + 1 = 2x +1 is an odd number.

Two odd numbers would be 2m+1 and 2n+1

Adding them gives:

2m+1 + 2n + 1 = 2m + 2n + 2 = 2(m + n + 1) = 2p where p = m + n + 1

Thus adding two odd numbers results in an even number.

Similarly for one even and one odd number.

An even number would be 2m and an odd number would be 2n+1

Adding them gives:

2m + 2n + 1 = 2(m + n) + 1 = 2p + 1 where p = m + n

Thus adding an even number and an odd number results in an odd number.