Divisibility Rules Calculator
Enter a number and this checks it against every divisibility rule up to eleven, showing which apply and why.
Divides by
2, 3, 4, 6, 8, 9
- Divisible by 2
- Yes
- Divisible by 3
- Yes
- Divisible by 4
- Yes
- Divisible by 5
- No
- Divisible by 6
- Yes
- Divisible by 8
- Yes
- Divisible by 9
- Yes
- Divisible by 10
- No
- Divisible by 11
- No
- By 2: The last digit is even.
- By 3: The digits add up to 18, which is a multiple of 3.
- By 4: The last two digits make 72, which is a multiple of 4.
- By 6: It has to pass both the 2 rule and the 3 rule.
- By 8: The last three digits are a multiple of 8.
- By 9: The digits add up to 18, which is a multiple of 9.
- The rule for 3 works because 10 leaves a remainder of 1 when divided by 3, and so does every power of 10. That makes a number worth the same as its digits added up, as far as 3 is concerned. The same argument gives the rule for 9.
About divisibility rules
A divisibility rule is a shortcut for telling whether one number divides into another exactly, without doing the division. Most of them look at only a digit or two, which makes them far faster than dividing and possible to do in your head.
Some are obvious once stated. A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, and by 10 if it ends in 0. Those follow directly from how place value works: every column above the ones is already a multiple of 10.
The interesting ones are 3 and 9, where you add the digits up. That works because 10 leaves a remainder of 1 when divided by 3 or 9, and so does every power of 10 — so as far as 3 and 9 are concerned, a number is worth the same as its digits added together. It is not a coincidence and it is not arbitrary.
The rules also combine. A number divides by 6 exactly when it divides by both 2 and 3, because 2 and 3 are the primes 6 is made of. The same trick gives a rule for 12, 15 or any other number whose prime factors you already have rules for — as long as those factors share nothing between them.
What it works out
- Rules for 2, 3, 4, 5, 6, 8, 9, 10 and 11
- Which apply to a given number, and why
- How the digit-sum rules work
- Combining rules for larger divisors
The formula
2: last digit even 3: digits add to a multiple of 3 4: last two digits 5: ends 0 or 5 9: digits add to a multiple of 9 11: alternating sum
Take 3,672. Its last digit is 2, so it divides by 2. Its digits add to 3 + 6 + 7 + 2 = 18, which is a multiple of both 3 and 9, so it divides by both. Its last two digits make 72, which is a multiple of 4. It passes 2 and 3, so it passes 6. It does not end in 0 or 5, so 5 and 10 are out.
The digit-sum rule is worth understanding rather than memorising. Write 3,672 as 3×1000 + 6×100 + 7×10 + 2. Now note that 1000, 100 and 10 each leave a remainder of 1 when divided by 9. So the whole number leaves the same remainder as 3 + 6 + 7 + 2 does. Add the digits and you have replaced the number with something much smaller that behaves identically.
The same argument works for 3, since 10 also leaves a remainder of 1 when divided by 3.
Eleven works the other way. Ten leaves a remainder of −1 when divided by 11, so the powers of ten alternate between +1 and −1, which is why you add and subtract the digits in turn rather than adding them all.
For anything bigger, combine. Twelve is 4 × 3, and 4 and 3 share no factors, so a number divides by 12 exactly when it passes both rules.
- Digit sum
- All the digits added together. Used for 3 and 9, and it can be repeated if still too large.
- Alternating sum
- Digits added and subtracted in turn from the right. Used for 11.
- Last two digits
- Enough for 4, because 100 is already a multiple of 4.
A worked example
Checking 3,672 against every rule.
That works out to 2, 3, 4, 6, 8, 9 .
- Divisible by 2
- Yes
- Divisible by 3
- Yes
- Divisible by 4
- Yes
- Divisible by 5
- No
- Divisible by 6
- Yes
- Divisible by 8
- Yes
- Divisible by 9
- Yes
- Divisible by 10
- No
- Divisible by 11
- No
Questions
How do I know if a number is divisible by 3?
Add its digits. If the total is a multiple of 3, so is the number. 3,672 gives 3 + 6 + 7 + 2 = 18, which is a multiple of 3.
Why does adding the digits work for 3 and 9?
Because 10, 100, 1,000 and every other power of ten leave a remainder of 1 when divided by 3 or 9. That makes a number leave the same remainder as its digits added up, so the two are interchangeable for this test.
What is the rule for 4?
Look at the last two digits only. If they form a multiple of 4, so does the whole number — because every hundred is already a multiple of 4. In 3,672 the 72 is a multiple of 4.
What is the rule for 8?
The last three digits, for the same reason: every thousand is a multiple of 8. It is less useful than the others because checking three digits is close to the work of dividing.
What is the rule for 11?
Add and subtract the digits alternately from the right. If the result is 0 or a multiple of 11, so is the number. For 2,728: 8 − 2 + 7 − 2 = 11.
Is there a rule for 7?
There are several, and none is quicker than dividing. That is why 7 is left out of most lists — for that one, just divide.
How do I check divisibility by 12 or 15?
Combine the rules of their prime factors. Twelve is 4 × 3, so check both. Fifteen is 3 × 5. The factors must share nothing between them — checking 2 and 6 for 12 does not work, since both already contain a 2.
What are divisibility rules for?
Simplifying fractions, finding factors, and spotting quickly whether a division will come out exactly. They are the fastest way into prime factorisation, since they tell you which prime to try next.