Maths
What are the divisibility rules from 2 to 12?
Quick answer
Divisibility rules are shortcuts that tell you whether a number divides exactly by 2, 3, 4, 5, 6, 8, 9, 10, 11 or 12 without doing the division, by looking at its last digits or its digit sum.
In one sentence
A divisibility rule is a quick test that tells you whether one whole number divides exactly into another, with no remainder, without doing the whole division.
Explained
Say you have 234 cookies and want to pack them into boxes of 9 with none left over. You could do the division, but there is a faster way: add the digits. 2 + 3 + 4 = 9, and 9 is divisible by 9, so 234 is too. (It makes 26 boxes.) That shortcut is a divisibility rule.
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"Divisible by" means "divides exactly, with remainder 0". 234 is divisible by 9 because 234 ÷ 9 = 26 exactly. It is not divisible by 5, because 234 ÷ 5 leaves a remainder of 4.
The rules come in three families, which makes them easier to remember:
- Look at the last digits. 2, 5 and 10 only need the last digit, because 10 is divisible by all three. 4 and 25 need the last two digits, because 100 is divisible by 4 and 25. 8 needs the last three digits, because 1000 is divisible by 8.
- Add up the digits. 3 and 9 use the digit sum. This works because 10, 100, 1000 and so on are each one more than a multiple of 9 (9, 99, 999), so a number and its digit sum leave the same remainder when divided by 3 or 9.
- Combine two rules. 6 = 2 × 3, so check 2 and 3. 12 = 3 × 4, so check 3 and 4. This only works when the two numbers share no common factor.
Then there are two special ones: 11 (alternate adding and subtracting the digits) and 7 (double the last digit and subtract it from the rest). They take a little more practice but save a lot of long division.
Divisibility rules help with factors and multiples, simplifying fractions and spotting prime numbers: a number that passes a test for 2, 3, 5, 7 or 11 (and is not that number itself) cannot be prime.
Examples
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even: 0, 2, 4, 6 or 8 | 846 ends in 6, so yes |
| 3 | Digit sum is divisible by 3 | 471: 4 + 7 + 1 = 12, so yes |
| 4 | Last two digits make a number divisible by 4 | 1316: 16 ÷ 4 = 4, so yes |
| 5 | Last digit is 0 or 5 | 735 ends in 5, so yes |
| 6 | Divisible by 2 and by 3 | 618: even, and 6 + 1 + 8 = 15, so yes |
| 7 | Double the last digit, subtract from the rest; the result is divisible by 7 | 343: 34 − 6 = 28, so yes |
| 8 | Last three digits make a number divisible by 8 | 7128: 128 ÷ 8 = 16, so yes |
| 9 | Digit sum is divisible by 9 | 5832: 5 + 8 + 3 + 2 = 18, so yes |
| 10 | Last digit is 0 | 3150 ends in 0, so yes |
| 11 | Alternating sum of the digits is divisible by 11 (0 counts) | 2728: 8 − 2 + 7 − 2 = 11, so yes |
| 12 | Divisible by 3 and by 4 | 1452: 1 + 4 + 5 + 2 = 12 and 52 ÷ 4 = 13, so yes |
| 25 | Last two digits are 00, 25, 50 or 75 | 1375 ends in 75, so yes |
Worked example: is 2401 divisible by 7? Take off the last digit (1), double it (2) and subtract from what is left: 240 − 2 = 238. Still big, so do it again: 23 − 16 = 7. That is divisible by 7, so 2401 is divisible by 7 (2401 = 7 × 343).
Common mistakes
- Using the last digit for 3 or 9. 13 ends in 3 but is not divisible by 3. For 3 and 9 you must add all the digits.
- Combining the wrong pair. Divisible by 2 and 4 does not mean divisible by 8: 12 passes both but 12 ÷ 8 is not exact. Only combine numbers with no common factor: 12 is 3 and 4 (not 2 and 6), and 8 needs its own rule.
- Using the digit sum for 2. For 2, only the last digit matters. 11 has an even digit sum (2) but is odd; 12 has an odd digit sum (3) but is even.
- Forgetting that 0 counts in the 11 rule. If the alternating sum is 0, the number is divisible by 11, as with 1001 or 5522.
Step by step
Step by step
- Check the last digitIt tells you about 2, 5 and 10 straight away.
- Add the digitsThe digit sum tells you about 3 and 9. If the sum is still big, add its digits again.
- Check the last two or three digitsLast two for 4 and 25, last three for 8.
- CombinePassed 2 and 3? Then 6. Passed 3 and 4? Then 12.
- Use the special tests if you need themAlternating sum for 11; “double the last digit and subtract” for 7.
When you need all the factors of a number, run through the tests in this order and note every one it passes. It is the quickest start to finding factor pairs or a prime factorisation.
Practice questions
Use the divisibility rules
Try each one, then check your answer.
- Is 846 divisible by 2?
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It ends in 6, which is even: yes (846 ÷ 2 = 423) - Is 735 divisible by 5? Is it divisible by 10?
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It ends in 5: yes by 5, no by 10 (it does not end in 0) - Is 471 divisible by 3?
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4 + 7 + 1 = 12, divisible by 3: yes (471 ÷ 3 = 157) - Is 5832 divisible by 9?
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5 + 8 + 3 + 2 = 18, divisible by 9: yes (5832 ÷ 9 = 648) - Is 1316 divisible by 4?
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The last two digits are 16, and 16 ÷ 4 = 4: yes (1316 ÷ 4 = 329) - Is 2734 divisible by 4?
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The last two digits are 34, and 34 ÷ 4 is not exact: no - Is 618 divisible by 6?
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Even, and 6 + 1 + 8 = 15 is divisible by 3: yes (618 ÷ 6 = 103) - Is 538 divisible by 6?
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Even, but 5 + 3 + 8 = 16 is not divisible by 3: no - Is 7128 divisible by 8?
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The last three digits are 128, and 128 ÷ 8 = 16: yes (7128 ÷ 8 = 891) - Is 343 divisible by 7?
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34 − 2 × 3 = 28, divisible by 7: yes (343 = 7 × 49) - Is 512 divisible by 7?
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51 − 2 × 2 = 47, not divisible by 7: no - Is 2728 divisible by 11?
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8 − 2 + 7 − 2 = 11: yes (2728 = 11 × 248) - Is 9163 divisible by 11?
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3 − 6 + 1 − 9 = −11, divisible by 11: yes (9163 = 11 × 833) - Which of 2, 3, 4, 5, 6, 8, 9 and 10 divide 360?
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Ends in 0 (2, 5, 10); digit sum 9 (3, 9); 60 ÷ 4 = 15 (4); 360 ÷ 8 = 45 (8); 2 and 3 (6): all of them - Which digit goes in the gap so that 4_2 is divisible by 9?
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4 + 2 = 6, and the digit sum must be 9: 3 (432 = 9 × 48) - What is the smallest digit that makes 73_ divisible by 6?
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It must be even, and 7 + 3 + d must be divisible by 3, so 10 + d = 12: 2 (732 = 6 × 122)
Word problems
Divisibility in real life
Try each one, then check your answer.
- Can 234 pupils be split into 9 equal teams?
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2 + 3 + 4 = 9, so yes: 26 pupils per team - A farmer has 412 eggs. Can she pack them all into boxes of 6?
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412 is even, but 4 + 1 + 2 = 7 is not divisible by 3, so no (there would be 4 eggs left over) - A year is usually a leap year if it is divisible by 4. Is 2028 a leap year? Is 2026?
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28 ÷ 4 = 7, so 2028 is; 26 ÷ 4 is not exact, so 2026 is not. (Century years must be divisible by 400: 2000 was a leap year, 1900 was not.) - Five friends win $135. Can they share it equally in whole dollars?
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135 ends in 5, so yes: $27 each - A shop has 1452 sweets to put in bags of 12. Will any be left over?
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1 + 4 + 5 + 2 = 12 (divisible by 3) and 52 ÷ 4 = 13 (divisible by 4), so none left over: 121 bags
Questions people ask
What are the divisibility rules?
They are shortcuts for checking whether a number divides exactly by another. The main ones are: 2 (even last digit), 3 (digit sum divisible by 3), 4 (last two digits), 5 (ends in 0 or 5), 6 (2 and 3), 8 (last three digits), 9 (digit sum divisible by 9), 10 (ends in 0) and 11 (alternating digit sum).
How many divisibility rules are there?
There is no fixed number: every whole number has some test. Schools usually teach the rules for 2, 3, 4, 5, 6, 8, 9 and 10, often adding 7, 11 and 12. Every number is divisible by 1, so that one needs no rule.
What is the divisibility rule for 7?
Double the last digit and subtract it from the number made by the other digits. If the result is divisible by 7 (including 0 or a negative multiple), so is the original. For 203: 20 − 6 = 14, so 203 is divisible by 7.
What is the divisibility rule for 11?
Starting from the right, subtract and add the digits in turn. If the result is 0 or a multiple of 11, the number is divisible by 11. For 2728: 8 − 2 + 7 − 2 = 11, so yes.
How do you remember the divisibility rules?
Group them: last digit for 2, 5 and 10; last two digits for 4 and 25; last three digits for 8; digit sum for 3 and 9; combinations for 6 and 12. An anchor chart on the wall with one example per rule helps a lot.
Why does the rule for 3 work?
Every place value is a multiple of 3 plus 1: 10 = 9 + 1, 100 = 99 + 1, and so on. So 471 = 4 × 99 + 7 × 9 + (4 + 7 + 1). The first two parts are multiples of 3, so 471 is divisible by 3 exactly when 4 + 7 + 1 is.
Is there a divisibility rule for 13?
Yes: take the last digit, multiply it by 4 and add it to the rest. If the result is divisible by 13, so is the number. For 182: 18 + 8 = 26 = 2 × 13, so 182 is divisible by 13.
Quick quiz
Divisibility rules
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Question 1 of 5
Which of these numbers is divisible by 3?
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Related topics
Factors and multiples
Factors are numbers that divide exactly into a number (the factors of 12 are 1, 2, 3, 4, 6 and 12). Multiples are the numbers in its times table (12, 24, 36…).
Prime numbers
A prime number is a whole number greater than 1 that has exactly two factors, 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13.
Division
Division is sharing into equal groups, or finding how many groups fit. The sign is ÷, so 12 ÷ 3 = 4. The answer is called the quotient.
Prime factorisation
Prime factorisation means writing a number as a product of prime numbers. For example, 60 = 2 × 2 × 3 × 5, which can be written as 2² × 3 × 5.
Dividing decimals
To divide by a whole number, divide as usual and put the decimal point straight above. To divide by a decimal, multiply both numbers by 10, 100… until the divisor is whole: 7.2 ÷ 0.6 = 72 ÷ 6 = 12.
Long division
Long division splits a big number into equal groups by repeating four steps: divide, multiply, subtract, bring down. Work from the left, one digit at a time.