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 \(2k + 1\)).
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The parity of an integer records which of two classes it belongs to: an even number is divisible by , and an odd number is not. Euclidean division gives the unique expressionThe remainder encodes the parity. This applies to negative numbers too: is odd, while is even. Zero is an even number.
Reduction modulo respects addition and multiplication, so the quotient ring records parity arithmetic. If have remainders , then has remainder , and has remainder . In particular, the sum of two odd integers is even, whereas their product is odd. The relation shows that changing sign preserves parity.