This cheat sheet covers the most useful divisibility rules for whole numbers. Students use these rules to tell whether a number can be divided evenly without doing long division. The rules help with factoring, simplifying fractions, finding common multiples, and checking answers.
A clean reference makes it easier to choose the right test quickly.
The main ideas are grouped into last-digit rules, digit-sum rules, and combined or special rules. Numbers divisible by , , and can be tested by the last digit, while and use the sum of the digits. Rules for and use the last or digits, and the rule for combines divisibility by and .
These patterns work because of how place value is built from powers of .
Key Facts
- A whole number is divisible by if its last digit is , , , , or .
- A whole number is divisible by if its last digit is or .
- A whole number is divisible by if its last digit is .
- A whole number is divisible by if the sum of its digits is divisible by .
- A whole number is divisible by if the sum of its digits is divisible by .
- A whole number is divisible by if the number formed by its last digits is divisible by .
- A whole number is divisible by if the number formed by its last digits is divisible by .
- A whole number is divisible by if it is divisible by both and .
Vocabulary
- Divisible
- A number is divisible by another number if the division has no remainder.
- Factor
- A factor is a whole number that divides another whole number evenly.
- Multiple
- A multiple is the product of a number and a whole number, such as being a multiple of .
- Remainder
- A remainder is the amount left over when one whole number does not divide another evenly.
- Digit Sum
- A digit sum is the result of adding all the digits in a number, such as for .
- Last Digit
- The last digit is the digit in the ones place of a whole number.
Common Mistakes to Avoid
- Using the last digit rule for or is wrong because divisibility by and depends on the sum of all digits, not the ones place.
- Checking only divisibility by for the rule for is wrong because a number must be divisible by both and to be divisible by .
- Using the last digit only for is wrong because the rule for uses the number formed by the last digits.
- Using the last digits for is wrong because the rule for uses the number formed by the last digits.
- Forgetting that can be a last digit in rules for , , and is wrong because numbers ending in are divisible by all three of those numbers.
Practice Questions
- 1 Is divisible by , , , , , or ? Show which rules you used.
- 2 Use divisibility rules to decide whether is divisible by , , , and .
- 3 Find a -digit number that is divisible by both and , and explain how you checked it.
- 4 Explain why a number that is divisible by must also be divisible by and .
Understanding Divisibility Rules Reference
Divisibility rules come from place value. In our base ten number system, each place to the left is worth ten times the place before it. This creates useful leftovers when numbers are grouped.
For example, tens, hundreds, and thousands each leave no remainder when divided by two, five, or ten if the final digit already meets the needed condition. That is why digits farther left do not affect those tests. For divisibility by four, every full hundred can be split into groups of four.
For divisibility by eight, every full thousand can be split into groups of eight. Only the final part of the number needs checking.
The digit sum patterns for three and nine have a different reason. Ten is one more than a multiple of nine and one more than a multiple of three. A number such as four hundred thirty two can therefore be treated, for these tests, like four plus three plus two.
Both forms give the same remainder after grouping by three or by nine. This is not a shortcut based on luck.
It is a result of the way decimal place values behave. Students do not need to prove this every time, but knowing the reason helps the rule feel less like a fact to memorize.
Combined tests need careful thinking. A number that passes the test for two and the test for three is divisible by six because six is made from two times three. It is important that both conditions are true.
Passing only one condition is not enough. Similar thinking appears when working with twelve, which needs divisibility by three and four. However, combined rules cannot always be made by simply joining any two tests.
For example, divisibility by four already includes divisibility by two, so checking both does not prove divisibility by eight. Pay attention to which factors are already contained inside another factor.
These skills appear in fraction work more often than students expect. Before reducing a fraction, divisibility checks can reveal common factors quickly. They help when sorting objects into equal groups, planning rows of seats, finding package sizes, and checking whether a result from multiplication or division makes sense.
When practicing, write the relevant digits clearly and do not confuse the last two digits with their digit sum. For a large number, ignore leading digits only when the rule allows it. A reliable habit is to name the divisor first, choose the matching test, then verify the conclusion with a quick division when the answer matters.