#a proof

For any two odd numbers x and y, we can represent them as:

x = 2k+1 y = 2r+1

Which then means:

x + y = (2k + 1) + (2r + 1)

= 2k + 2r + 2

= 2(k + r + 1)

Which is divisible by 2, thus even. Similarly, or any two even numbers x and y, we can represent them as:

x = 2k y = 2r

Which then means:

x + y = 2k + 2r

= 2(k + r)

Which is also divisible by 2, thus even. Finally, for an odd number x and an even number y, we can represent them as:

x = 2k + 1 y = 2r

Which then means:

x + y = (2k + 1) + 2r

= 2k + 2r + 1 = 2(k + r) + 1

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