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 . An even number has parity because the remainder after dividing by is , while an odd number has parity because the remainder after dividing by is .
Here are a few arithmetic rules of parity that are extremely useful:
even even = even
even odd= odd
even odd= even
odd 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. If is an integer, what is the parity of ?
Solution: Since is an integer, is also an integer. Thus, shows that the parity of is .
2. If are integers, what is the parity of ?
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)
3. If is an integer, what is the parity of ?
Solution: . Note that have different parity. Hence, by the arithmetic rules of parity, the parity of is .