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 2. An even number has parity 00 because the remainder after dividing by 22 is 00, while an odd number has parity 11 because the remainder after dividing by 22 is 11.

Here are a few arithmetic rules of parity that are extremely useful:

even ± \pmeven = even

odd± \pmodd=even

even ± \pmodd= odd

even× \timeseven= even

even ×\timesodd= even

odd × \timesodd= 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. Worked Examples

1. If  nn is an integer, what is the parity of 2n+22n+2?

Solution: Since nn is an integer, n+1n+1 is also an integer. Thus, 2n+2=2(n+1)+02n+2 = 2(n+1) + 0 shows that the parity of 2n+22n+2 is 00.

 

2. If a,ba, b are integers, what is the parity of a×ba \times b?

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) Parity of a× Parity of b= Parity of ab\mbox{Parity of } a \times \mbox{ Parity of } b = \mbox{ Parity of } ab

&nsbp;

3. If kk is an integer, what is the parity of k2+k k^2 + k?

Solution: k2+k=k(k+1) k^2 + k = k (k+1). Note that k,(k+1) k, (k+1) have different parity. Hence, by the arithmetic rules of parity, the parity of k(k+1) k(k+1) is 0 0.

Note by Arron Kau
5 years, 2 months ago

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Well said!

lalitha sree - 5 years, 2 months ago

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Nice..

Munendra Kumar - 4 years, 11 months ago

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good..........

Shatrughna Kumar - 4 years, 10 months ago

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excellent !

Tushar Malik - 4 years, 10 months ago

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niceee ! I liked this

Rohan Chandra - 5 years ago

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