 Parity (mathematics)

The parity of an integer records which of two classes it belongs to: an even number is divisible by $2$, and an odd number is not. Euclidean division gives the unique expressionThe remainder $r=n\mathrm{mod}2$ encodes the parity. This applies to negative numbers too: $-3=2(-2)+1$ is odd, while $-4=2(-2)$ is even. Zero is an even number.

Reduction modulo $2$ respects addition and multiplication, so the quotient ring $\mathbb{Z}/2\mathbb{Z}$ records parity arithmetic. If $a,b$ have remainders $r,s$, then $a+b$ has remainder $(r+s)\mathrm{mod}2$, and $ab$ has remainder $rs$. In particular, the sum of two odd integers is even, whereas their product is odd. The relation $-n\equiv n(\mathrm{mod}2)$ shows that changing sign preserves parity.

In mathematics, "parity" refers to the property of an integer's evenness or oddness. Specifically, an integer is said to be **even** if it is divisible by 2 (i.e., it can be expressed in the form \(2k\) where \(k\) is an integer), and it is **odd** if it is not divisible by 2 (i.e., it can be expressed in the form \(2k + 1\)).