# Parity of Integers

#### Contents

## 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.

## Problem Solving

**Parity** appears in problem solving and can prove basic,but interesting results:

## If John has

rnumber of boxes of chocolates, whereris an odd natural number,and each box containedmnumber of chocolates where \(m=2n\) for some natural \(n\).John's friend Alex stole

wnumber of boxes from the set of boxes and in hurry,put a chocolate from the stolen box to some other unstolen box.He thought that John would not be able to blame him asrwas quite large.(and probably not notice the difference in number of boxes) However,John mathematically pointed out that something was wrong and immediately pointed out his mischievous friend Alex. Can you tell how?Trick question? Not quite! We have a very powerful tool in math,

Parity.Initially, John had an

oddnumber of boxes.Each box contained anevennumber of chocolates. So the total number of chocolates must be even(EVEN+EVEN+EVEN.........odd times=EVEN). We don't know ifwis even or odd.But that does not matter. Even or Odd number of boxes stolen,resulted inevennumber of chocolates remaining(EVEN+EVEN+EVEN.........wtimes(that may be odd or even)=Even). So even number of chocolates were stolen.But, in hurry, he did the biggest mistake,thus giving the chance to John to make out the thief.From his(Alex's) stolen boxes,by mistake,he put 1 chocolate from his box to the unstolen box,thus, making the parity of the final left out chocolates

odd(that is,1),opposite from the parity that the number of chocolates initially had(0,that is,even).\(\textbf{This way he spotted out the difference}\).

**Cite as:**Parity of Integers.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/modular-arithmetic-parity/