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What are the Divisibility Rules?

Divisibility rules are simple shortcuts that help us determine if one number is divisible by another without performing long division. These guidelines apply to numbers like 2, 3, 5, and 10, and involve checking a number's digits. Understanding these rules can make math much easier. Ready to unlock the secrets of quick mental math? Let's examine these rules together.
Tricia Christensen
Tricia Christensen
Tricia Christensen
Tricia Christensen

Divisibility rules can be simple and easy to remember tests you can perform on a number to determine if it will be evenly divided by another number. Some of these rules are quick to memorize and you probably already know some of them. For example, if the last digit of a number is even, you’re likely to be aware that the number can be evenly divided by 2. Another of the divisibility rules most people may know without thinking is that numbers ending in a 0 will always be divisible by 10 and by 5.

You can apply the following divisibility rules to numbers to help you determine whether you’ll get an even result:

A number will be divisible by 3 if the sum of the digits is divisible by 3.
Example: 228 is evenly divisible by 3 because 2 + 2 + 8 = 12 and 12 is divisible by 3.

4 will evenly divide a number if the last two digits of that number are divisible by 4.
Example: 788 is divisible by 4 because 88 is divisible by 4.

Any number ending in 0 or 5 will be evenly divided by 5, and 10 will evenly divide any number ending in 0.

If a number is divisible by 2 and 3, it is also divisible by 6.
Example: 180/2 = 90 and 180/3 = 60. Therefore 6 will also divide 180 evenly with a result of 30.

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Woman holding a book

When the sum of a number’s digits equals a number divisible by 9, that number will always be divisible by nine.
Example: The number 621 has a digit sum of 9. 9 will divide 621 evenly with a result of 69.
You can take this divisibility rules for 9 to determine if 18 will evenly divide numbers. If both 2 and 9 will divide a number, 18 will also divide it.

The above examples are probably the easiest of the divisibility rules to remember. Others get significantly more complicated and may involved multiple manipulations of a number prior to deciding whether it can be evenly divided by a divisor. Sometimes it takes less time to simply do division than it does to apply one of the divisibility rules to a number, and these rules exist for very large numbers too. With complicated operations you can determine things like whether 71 or 79 will evenly divide other numbers.

The divisibility rules for 8 and 7 fall into this more complicated arena. For some math applications they may be useful. However, with smaller numbers you may simply want to do the division to determine if 8 or 7 are factors of these numbers.

Tricia Christensen
Tricia Christensen

Tricia has a Literature degree from Sonoma State University and has been a frequent AllTheScience contributor for many years. She is especially passionate about reading and writing, although her other interests include medicine, art, film, history, politics, ethics, and religion. Tricia lives in Northern California and is currently working on her first novel.

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Tricia Christensen
Tricia Christensen

Tricia has a Literature degree from Sonoma State University and has been a frequent AllTheScience contributor for many years. She is especially passionate about reading and writing, although her other interests include medicine, art, film, history, politics, ethics, and religion. Tricia lives in Northern California and is currently working on her first novel.

Learn more...

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