Divisibility Rules, Tests and Shortcuts
Divisibility Rules determine whether a number is exactly divisible by another number without performing complete division. This page explains divisibility tests for common numbers such as 2, 3, 4, 5, 6, 7, 8, 9, 11 and 13, along with formulas, shortcuts and numerical examples.
What Are Divisibility Rules?
For example, 348 is divisible by 4 because its last two digits, 48, are divisible by 4. It is also divisible by 3 because the sum of its digits is 3 + 4 + 8 = 15, which is divisible by 3. A number may satisfy more than one divisibility rule.
Divisibility Rules Formula & Tricks
Important Formulas
The quotient k must be an integer, so the remainder after division by d is zero.
The rule applies repeatedly to the digit sum if necessary. For 729, the digit sum is 18, so 729 is divisible by both 3 and 9.
For 2728, the difference between (2 + 2) and (7 + 8) is |4 − 15| = 11, so 2728 is divisible by 11.
Repeat the process until the result is easy to test. If the final result is divisible by 7, the original number is divisible by 7.
Quick Tricks
For relatively prime factors, a number is divisible by their product when it is divisible by each factor. For example, divisibility by 6 requires divisibility by both 2 and 3.
Rules for powers of 2 and 5 depend only on the final digits. Check the last two digits for 4, the last three digits for 8, and the last two digits for 25.
For tests of 3 and 9, a large digit sum can be reduced again without changing divisibility.
For a large number, remove the last digit, double it, and subtract the result from the remaining number. The sign may be reversed if the resulting number is easier to test.
Divisibility Rules Concepts
Divisibility Rules for 2, 5 and 10
These tests use the units digit because 10, 100, 1000 and higher place values are divisible by 2, 5 and 10 as appropriate. The other digits do not affect the remainder in these cases.
Divisibility Rules for 3 and 9
For 84,615, the digit sum is 8 + 4 + 6 + 1 + 5 = 24. Since 24 is divisible by 3 but not by 9, the number is divisible by 3 but not by 9.
Divisibility Rules for 4, 8 and 16
This works because every place value before the checked digits is a multiple of 4, 8 or 16. For 53,216, the last two digits 16 are divisible by 4, the last three digits 216 are divisible by 8, and the last four digits 3216 are divisible by 16.
Divisibility Rules for 6, 12, 15, 18 and 20
The conditions must all hold at the same time. For divisibility by 12, the digit sum must be divisible by 3 and the last two digits must be divisible by 4. For divisibility by 18, the number must be even and its digit sum must be divisible by 9.
Divisibility Rules for 7, 11 and 13
For 7, 203 gives 20 − 2 × 3 = 14, which is divisible by 7, so 203 is divisible by 7. For 11, the difference between alternate digit sums must be 0 or a multiple of 11. For 13, 286 gives 28 + 4 × 6 = 52, which is divisible by 13, so 286 is divisible by 13.
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Divisibility Rules for 2, 4 and 8
Learn how to test whether a number is divisible by 2, 4, or 8 using the relevant digit and last-digit rules, with clear applications for aptitude questions.
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Divisibility Rules: Quick Revision
Use the final digits, digit sums or factor conditions according to the divisor.
- 2: The last digit is 0, 2, 4, 6 or 8.
- 3: The sum of digits is divisible by 3.
- 4: The last two digits are divisible by 4.
- 5: The last digit is 0 or 5.
- 6: The number is divisible by both 2 and 3.
- 7: Subtract twice the last digit from the remaining number.
- 8: The last three digits are divisible by 8.
- 9: The sum of digits is divisible by 9. 10: The last digit is 0. 11: The alternating digit sum is 0 or a multiple of 11. 12: The number is divisible by both 3 and 4. 13: Add four times the last digit to the remaining number.
Divisibility Rules FAQs
What is the divisibility rule for 12?
A number is divisible by 12 if it is divisible by both 3 and 4. Its digit sum must be divisible by 3, and its last two digits must be divisible by 4.
Is every number divisible by 6 also divisible by 3?
Yes. Since 6 = 2 × 3, divisibility by 6 requires divisibility by both 2 and 3. For example, 246 is divisible by 6 and its digit sum, 12, is divisible by 3.
How can divisibility by 11 be checked?
Find the difference between the sums of alternate digits. If the absolute difference is 0 or a multiple of 11, the number is divisible by 11. For 5,214, |(5 + 1) − (2 + 4)| = 0.
What is the divisibility test for 7?
Double the last digit and subtract it from the number formed by the remaining digits. For 161, 16 − 2 × 1 = 14, so 161 is divisible by 7.
How is divisibility by 13 tested?
Add four times the last digit to the remaining leading part. For 286, 28 + 4 × 6 = 52, and 52 is divisible by 13; therefore, 286 is divisible by 13.
What is the difference between the tests for 4 and 8?
For 4, check the last two digits. For 8, check the last three digits. For example, 1,124 is divisible by 4 because 24 is divisible by 4, but it is not divisible by 8 because 124 is not divisible by 8.