Parity

Definition

Parity is a term we use to express if a given integer is even or odd. The parity of a number depends only on its remainder after dividing by $2$. An even number has parity $0$ because the remainder after dividing by $2$ is $0$, while an odd number has parity $1$ because the remainder after dividing by $2$ is $1$.

Here are a few arithmetic rules of parity that are extremely useful:

even $\pm$even = even

odd$\pm$odd=even

even $\pm$odd= odd

even$\times$even= even

even $\times$odd= even

odd $\times$odd= odd

Parity is often useful for verifying whether an equality is true or false by using the parity rules of arithmetic to see whether both sides have the same parity.

1. Worked Examples

1. If  $n$ is an integer, what is the parity of $2n+2$?

Solution: Since $n$ is an integer, $n+1$ is also an integer. Thus, $2n+2 = 2(n+1) + 0$ shows that the parity of $2n+2$ is $0$.

2. If $a, b$ are integers, what is the parity of $a \times b$?

Solution: We know that an odd number multiplied by an odd number remains odd, an even number multiplied an odd number is even, and an even number multiplied by an even number is even. This can be summarized as (check for yourself) $\mbox{Parity of } a \times \mbox{ Parity of } b = \mbox{ Parity of } ab$

&nsbp;

3. If $k$ is an integer, what is the parity of $k^2 + k$?

Solution: $k^2 + k = k (k+1)$. Note that $k, (k+1)$ have different parity. Hence, by the arithmetic rules of parity, the parity of $k(k+1)$ is $0$.