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.
Two integers have opposite parity when one is an even number and the other is an odd number. Their sum is then an odd number.
An odd number is an integer not divisible by . Equivalently, it has the unique formThe Euclidean division remainder is one. Examples include ; zero is excluded because it is an even number. The property is independent of primality: is a prime number, is a composite number, and is neither.
An even number is an integer divisible by , so it has the form for an integer . Examples include ; zero is even because . This is the zero remainder class in Euclidean division by .
The only prime number that is even is . Every greater positive even integer has as a proper divisor. This explains why many primality testing procedures handle separately before discarding other even inputs.
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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 \(2k + 1\)).