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Parity - Even and Odd Integers



Parity is a topic that is only briefly touched upon in school.  It is one of those topics that students need extra study on in order to do better on the math portion of the SAT.

Parity just means whether an integer is even or odd.  It is a property of an integer.  For example, we say that 0 has even parity, and 1 has odd parity.

Even integers: ..., -4, -2, 0, 2, 4, ...
Odd integers: ..., -3, -1, 1, 3, ...

You should be able to tell quickly if an integer is even or odd.  Just look at the last digit.  If the last digit is even (0, 2, 4, 6, 8), the integer is even.  If the last digit is odd (1, 3, 5, 7, 9), the integer is odd.  When looking at the examples below, don't actually perform the operations!  Try to sense quickly whether the integers are even or odd, and if the result should be even or odd.

You need to know how parity behaves when two integers are added, subtracted, multiplied, or divided.  A special case of multiplication is taking powers.  In this post, I will go over the rules for these.  Problems can get harder when variables or other complications are involved--this may be covered in a future article.


Addition and Subtraction


The same rules hold for both addition and subtraction, so I will just show the ones for addition.

even + even = even
even + odd = odd
odd + odd = even

Put another way:

Sum of two integers with the same parity (both even or both odd) = even
Sum of two integers with different parity = odd

Examples:

24 + 88 = even
-2 + 34 = even
51 - 28 = odd


Multiplication


even * even = even
even * odd = even
odd * odd = odd

Put another way:

even * (any integer, even or odd) = even
odd * odd = odd (is only way to get odd result)

Examples:

2 * 9 = even
-2 * 6 = even
5 * 7 = odd
-5 * 3 = odd


Division


You need to be careful about division when it comes to integers and parity.

First of all, when you divide two integers, the result is not always an integer.  You can't tell if the result is an integer only by knowing the parity of the two integers you are dividing!

0 / 0 = undefined
0 / (non-zero integer) = 0

For the following, assume division gives an integer:

even / odd = even
even / even = could be even or odd
odd / odd = odd

Examples:

45 / 1 = 45 odd
45 / 3 = 15 odd
45 / 7 = not an integer

12 / 2 = 6 even
12 / 4 = 3 odd


Powers


even ^ (any positive integer not 0) = even
odd ^ (any positive integer) = odd

Focus on the base, not the exponent!  The result has the same parity as the base.

Examples:

2^4 = even
3^4 = odd
(-48)^23 = even
(-513)^4 = odd


Related facts


  • 2 is the only even prime number; all other prime numbers are odd


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