Parity of Integers
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.
If \(n\) is an integer, what is the parity of \(2n+2\)?
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 \(0\). \(_\square\)
If \(a, b\) are integers, what is the parity of \(a \times b\)?
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) \[\mbox{Parity of } a \times \mbox{ Parity of } b = \mbox{ Parity of } ab. \ _\square \]
If \(k\) is an integer, what is the parity of \( k^2 + k\)?
\( k^2 + k = k (k+1)\). Note that \( k, (k+1)\) have different parity. Hence, by the arithmetic rules of parity, the parity of \( k(k+1)\) is \( 0\). \(_\square\)
Do you know what's odd? Integers that aren't divisible by \(2\)! Another way of stating this is that a number \(n\) is odd iff \(n \equiv 1 \pmod2\). Integers congruent to \(0 \pmod2\) are called even.
In mathematical problem solving, parity is often useful in solving problems. Parity is whether an integer is even or odd. There are several properties of odd and even numbers that can be extremely useful in problem solving.
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