Is the sum of two odd numbers always odd?

Sep 21, 2015

The sum of two odd numbers is always even.

It can only be odd (too) if using modular arithmetic with an odd modulus.

Explanation:

If ${n}_{1}$ and ${n}_{2}$ are odd then $\exists {k}_{1} , {k}_{2}$ such that ${n}_{1} = 2 {k}_{1} + 1$ and ${n}_{2} = 2 {k}_{2} + 1$.

So we find:

${n}_{1} + {n}_{2} = \left(2 {k}_{1} + 1\right) + \left(2 {k}_{2} + 1\right) = 2 \left({k}_{1} + {k}_{2} + 1\right)$

which is a multiple of $2$ and therefore even.

In modular arithmetic with an odd modulus all numbers are both odd and even.