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Even and odd numbers

An even number can be written 2n2n, and an odd number can be written 2n+12n + 1. The sum of two odd numbers is then 2(m+n+1)2(m + n + 1), which is even. This is a direct proof: you write the numbers in general form and calculate. A concrete pair, for example 7 and 11, can be used as a check, but the proof is in the general calculation. The product of two odd numbers is, by contrast, odd.

(2m+1)+(2n+1)=2(m+n+1)(2m+1)+(2n+1) = 2(m+n+1)two odd numbers give an even number

Symbols

m, nm,\ nintegers

Example

7+11=187 + 11 = 18, and 18/2=918/2 = 9, so the sum is even. With m=3m = 3 and n=5n = 5 it is (2⋅3+1)+(2⋅5+1)(2\cdot 3 + 1) + (2\cdot 5 + 1).

Write an odd number as 2n+12n + 1, not as n+1n + 1, because n+1n + 1 can be either even or odd.
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← Direct proof

Part of Mathematics R1: Logic and proof.