Check the Last Digit
The last digit must be 0, 2, 4, 6, or 8.
746 ends in 6, so 746 is divisible by 2.
Free Math Guide
Divisibility Rules Explained
Learn how to decide whether a whole number is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 11, or 12 without performing long division.
A number is divisible by another number when the division produces a whole-number quotient with a remainder of zero. Divisibility rules let you test for that result by examining particular digits instead of completing the entire division.
What Does Divisible Mean?
If one whole number divides another evenly, there is no fractional part and no remainder. For example, 42 is divisible by 6 because 42 ÷ 6 equals the whole number 7.
42 ÷ 6 = 7 remainder 0
In contrast, 43 is not divisible by 6 because dividing 43 into groups of 6 leaves 1 behind.
Divisibility Is Exact
A quotient being close to a whole number is not enough. The remainder must be exactly zero.
Divisibility Rules for 2 Through 12
Each rule focuses on the smallest part of the number needed to make a reliable decision.
Add the Digits
The sum of the digits must be divisible by 3.
5 + 7 + 1 = 13, so 571 is not divisible by 3.
Check the Final Two Digits
The number formed by the final two digits must be divisible by 4.
3,516 ends in 16, so it is divisible by 4.
Look for 0 or 5
The last digit must be either 0 or 5.
2,735 ends in 5, so it is divisible by 5.
Combine the Rules for 2 and 3
The number must be divisible by both 2 and 3.
258 is even and 2 + 5 + 8 = 15, so it is divisible by 6.
Check the Final Three Digits
The number formed by the final three digits must be divisible by 8.
12,024 ends in 024, and 24 ÷ 8 = 3.
Add the Digits
The sum of the digits must be divisible by 9.
7 + 2 + 9 = 18, so 729 is divisible by 9.
Look for a Final Zero
The last digit must be 0.
4,230 ends in 0, so it is divisible by 10.
Use Alternating Digit Sums
Find the difference between the sums of alternating digits. The difference must be 0 or a multiple of 11.
For 5,148: (5 + 4) − (1 + 8) = 0.
Combine the Rules for 3 and 4
The number must be divisible by both 3 and 4.
1,332 has digit sum 9 and ends in 32, so it is divisible by 12.
Why Do Divisibility Rules Work?
The rules come from patterns in our base-ten number system. They are shortcuts, but they are not guesses.
Rules for 2, 5, and 10
The ones digit determines the remainder because every higher place value is a multiple of 10.
Rule for 4
Every multiple of 100 is divisible by 4, so only the final two digits can affect the remainder.
Rule for 8
Every multiple of 1,000 is divisible by 8, so only the final three digits matter.
Rules for 3 and 9
Powers of 10 leave the same remainder as 1 when divided by 3 or 9, which makes the digit sum equivalent for testing divisibility.
Rule for 11
Powers of 10 alternate between remainders of 1 and −1 when divided by 11, producing alternating digit sums.
Worked Divisibility Examples
Which Rules Apply to 378?
- The last digit is 8, so the rule for 2 applies.
- The digit sum is 3 + 7 + 8 = 18, so the rules for 3 and 9 apply.
- Because both 2 and 3 apply, the rule for 6 also applies.
- The final two digits, 78, are not divisible by 4.
378 is divisible by 2, 3, 6, and 9.
Which Rules Apply to 5,148?
- It ends in 8, so it is divisible by 2.
- The digit sum is 18, so it is divisible by 3 and 9.
- The final two digits are 48, so it is divisible by 4.
- It is divisible by 6 and 12 because the required combined rules apply.
- The alternating sums are 5 + 4 = 9 and 1 + 8 = 9, so it is divisible by 11.
5,148 is divisible by 2, 3, 4, 6, 9, 11, and 12.
Is 12,024 Divisible by 8?
Only the final three digits are needed.
024 = 24 and 24 ÷ 8 = 3
Yes. Therefore, 12,024 is divisible by 8.
Is 7,436 Divisible by 3?
Add every digit.
7 + 4 + 3 + 6 = 20
No. Since 20 is not divisible by 3, neither is 7,436.
Divisibility Rules for 6 and 12
The rules for 6 and 12 combine simpler tests. Every required condition must be true.
Divisible by 6
A number must be divisible by both 2 and 3.
234 is even, and 2 + 3 + 4 = 9. Therefore, 234 is divisible by 6.
Divisible by 12
A number must be divisible by both 3 and 4.
516 has digit sum 12 and ends in 16. Therefore, 516 is divisible by 12.
One Condition Is Not Enough
The number 22 is divisible by 2 but not by 3, so it is not divisible by 6. The number 124 is divisible by 4 but not by 3, so it is not divisible by 12.
Common Divisibility Mistakes
Using the Digit Sum for Every Rule
Digit sums test divisibility by 3 and 9. They do not directly test divisibility by 4, 5, or 8.
Checking Only the Last Digit for 4 or 8
Use the final two digits for 4 and the final three digits for 8.
Forgetting Both Parts of a Combined Rule
Divisibility by 6 requires both 2 and 3. Divisibility by 12 requires both 3 and 4.
Treating 0 as a Failed Difference for 11
A difference of 0 passes the rule for 11 because 0 is a multiple of 11.
Stopping After Finding One Rule
A number can satisfy several rules at once. Test every divisor requested by the problem.
How to Test a Number Efficiently
1. Last digit
Test the rules for 2, 5, and 10 immediately.
2. Digit sum
Add all the digits once and use the result for both 3 and 9.
3. Ending digits
Use the final two digits for 4 and the final three digits for 8.
4. Combined rules
Use your earlier conclusions to test 6 and 12 without starting over.
5. Alternating sums
If 11 is included, compare the two alternating digit sums.
Practice Divisibility Rules
Select every rule that applies to a generated whole number, receive immediate feedback, and review a clear explanation for every correct divisor.
Open the Divisibility Rules ToolRelated Math Resources
Interactive Divisibility Practice
Apply the rules to generated numbers and receive immediate explanations.
Prime Factorization Guide
Break composite numbers into prime factors using organized factor trees.
Greatest Common Factor Guide
Identify the greatest factor shared by two or more whole numbers.
Math Resources
Browse additional guides, worksheets, lessons, and interactive tools.