Divisibility Rules Reference Cheat Sheet

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 $2$ , $5$ , and $10$ can be tested by the last digit, while $3$ and $9$ use the sum of the digits. Rules for $4$ and $8$ use the last $2$ or $3$ digits, and the rule for $6$ combines divisibility by $2$ and $3$ .

These patterns work because of how place value is built from powers of $10$ .

Key Facts

A whole number is divisible by $2$ if its last digit is $0$ , $2$ , $4$ , $6$ , or $8$ .
A whole number is divisible by $5$ if its last digit is $0$ or $5$ .
A whole number is divisible by $10$ if its last digit is $0$ .
A whole number is divisible by $3$ if the sum of its digits is divisible by $3$ .
A whole number is divisible by $9$ if the sum of its digits is divisible by $9$ .
A whole number is divisible by $4$ if the number formed by its last $2$ digits is divisible by $4$ .
A whole number is divisible by $8$ if the number formed by its last $3$ digits is divisible by $8$ .
A whole number is divisible by $6$ if it is divisible by both $2$ and $3$ .

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 $24$ being a multiple of $6$ .
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 $4 + 5 + 6 = 15$ for $456$ .
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 $3$ or $9$ is wrong because divisibility by $3$ and $9$ depends on the sum of all digits, not the ones place.
Checking only divisibility by $2$ for the rule for $6$ is wrong because a number must be divisible by both $2$ and $3$ to be divisible by $6$ .
Using the last digit only for $4$ is wrong because the rule for $4$ uses the number formed by the last $2$ digits.
Using the last $2$ digits for $8$ is wrong because the rule for $8$ uses the number formed by the last $3$ digits.
Forgetting that $0$ can be a last digit in rules for $2$ , $5$ , and $10$ is wrong because numbers ending in $0$ are divisible by all three of those numbers.

Practice Questions

1 Is $7,428$ divisible by $2$ , $3$ , $4$ , $6$ , $8$ , or $9$ ? Show which rules you used.
2 Use divisibility rules to decide whether $5,670$ is divisible by $5$ , $6$ , $9$ , and $10$ .
3 Find a $3$ -digit number that is divisible by both $4$ and $9$ , and explain how you checked it.
4 Explain why a number that is divisible by $10$ must also be divisible by $2$ and $5$ .

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Divisibility Rules Reference Cheat Sheet

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