The rules at a glance
| Divisible by | Rule | Example | Because |
|---|---|---|---|
| 2 | The last digit is even (0, 2, 4, 6, 8) | 3,918 = 2 × 1,959 | last digit 8 |
| 3 | The sum of the digits is divisible by 3 | 4,731 = 3 × 1,577 | digit sum 15 |
| 4 | The number formed by the last two digits is divisible by 4 | 7,316 = 4 × 1,829 | last two digits 16 |
| 5 | The last digit is 0 or 5 | 2,485 = 5 × 497 | last digit 5 |
| 6 | Divisible by both 2 and 3 | 5,412 = 6 × 902 | even, digit sum 12 |
| 8 | The number formed by the last three digits is divisible by 8 | 9,104 = 8 × 1,138 | last three digits 104 = 8 × 13 |
| 9 | The sum of the digits is divisible by 9 | 8,874 = 9 × 986 | digit sum 27 |
| 10 | The last digit is 0 | 6,530 = 10 × 653 | last digit 0 |
| 11 | The alternating sum of the digits (from the right: +, −, +, …) is divisible by 11 | 9,427 = 11 × 857 | 7 − 2 + 4 − 9 = 0 |
| 12 | Divisible by both 3 and 4 | 3,516 = 12 × 293 | digit sum 15, last two digits 16 |
| 25 | The last two digits are 00, 25, 50 or 75 | 4,175 = 25 × 167 | last two digits 75 |
Every example in the table is divided out when this page is built, so a wrong example would stop the page from being published.
Rules that look at the last digits: 2, 4, 5, 8, 10, 25
Any whole number is some multiple of 10 plus its last digit. Since 10 is divisible by 2 and 5, only the last digit decides divisibility by 2, 5 and 10. In the same way 100 is divisible by 4 and 25, so only the last two digits matter for those, and 1,000 is divisible by 8, so only the last three matter for 8. That is why 7,316 is divisible by 4 the moment you see that 16 is.
Rules that use the digit sum: 3 and 9
Every power of ten is one more than a multiple of 9: 10 = 9 + 1, 100 = 99 + 1, 1,000 = 999 + 1. So a number and the sum of its digits leave the same remainder when divided by 9, and also by 3, which divides 9. Add the digits, and if the sum is still large, add its digits again: 8,874 gives 27, and 27 gives 9.
Combined rules: 6 and 12
When a divisor is the product of two numbers with no common factor, a number is divisible by it exactly when it is divisible by both. 6 = 2 × 3 and 12 = 3 × 4, so the tests combine. The trap is using factors that share a factor: divisible by 2 and by 4 does not make a number divisible by 8 (12 is the counter-example).
The awkward ones: 7 and 11
For 7: double the last digit and subtract it from the rest of the number; repeat. 2,401 → 240 − 2 = 238 → 23 − 16 = 7, a multiple of 7, so 2,401 is one too. It works because 21 is a multiple of 7, so subtracting twice the last digit is the same as subtracting a multiple of 7 and dividing by 10, neither of which changes divisibility by 7.
For 11: starting from the right, add and subtract the digits in turn. For 9,427 that is 7 − 2 + 4 − 9 = 0; zero counts as divisible, so 9,427 = 11 × 857. The reason is that 10 is one less than 11, so the powers of ten alternate between one more and one less than a multiple of 11.
Where the rules save time
- Reducing fractions: spot a common factor before reaching for the fraction calculator, which reduces any fraction exactly.
- Prime factors: test 2, 3 and 5 first and divide them out; the prime factorization calculator then finishes the job and shows each step.
- Common factors and multiples: the LCM and GCF calculator finds the greatest common factor directly, which is the number to cancel.
- Checking long division: if the dividend passes the rule for the divisor, the long division calculator should show a remainder of zero; if it does not, look for a slip.
- Splitting things evenly: 144 items split into 6, 8, 9 or 12 equal groups, which the rules show at a glance.
Frequently asked questions
How do you tell if a number is divisible by 3?
Add up its digits. If the sum is divisible by 3, so is the number. 4,731 has a digit sum of 15, which is divisible by 3, so 4,731 is too: 4,731 = 3 × 1,577. The rule works because 10 leaves a remainder of 1 when divided by 3, so every digit counts at face value.
What is the divisibility rule for 7?
Double the last digit and subtract it from the number formed by the remaining digits; repeat until the result is small. If that result is divisible by 7, so was the original. For 2,401: 240 − 2 = 238, then 23 − 16 = 7, which is divisible by 7, and indeed 2,401 = 7 × 343.
Why does the rule for 9 work?
Because 10 = 9 + 1, every power of ten is one more than a multiple of 9: 10, 100, 1,000 all leave a remainder of 1. So a number leaves the same remainder when divided by 9 as the sum of its digits does. The same reasoning gives the rule for 3.
Is 0 divisible by every number?
Yes. Zero divided by any nonzero number is zero with no remainder, so 0 is a multiple of every whole number. Dividing by zero, on the other hand, is undefined.
How do divisibility rules help with fractions?
They find common factors quickly. In 252/396 both digit sums are divisible by 9 and both numbers are even and pass the rule for 4, so you can cancel 36 at once: 252/396 = 7/11. The fraction calculator and the LCM and GCF calculator do the same reduction in one step.