Divisibility rules chart
A divisibility rule is a quick test that tells you whether one number divides evenly into another, without doing the division. A number is divisible by 2 if its last digit is even and by 3 if its digits add up to a multiple of 3.
Free printable PDF, 4th grade to 6th grade.
One page, black and white, prints the same every time.
What is on the sheet
| Divisible by | The rule | Example |
|---|---|---|
| 2 | The last digit is even: 0, 2, 4, 6 or 8. | 1,358 ends in 8. |
| 3 | The digits add up to a multiple of 3. | 471: 4 + 7 + 1 = 12 |
| 4 | The last two digits make a multiple of 4. | 2,316: 16 = 4 × 4 |
| 5 | The last digit is 0 or 5. | 845 ends in 5. |
| 6 | The number passes the tests for 2 and for 3. | 1,254: even, and 1 + 2 + 5 + 4 = 12 |
| 7 | Double the last digit and subtract it from the rest. The answer is a multiple of 7. | 343: 34 − 6 = 28 |
| 8 | The last three digits make a multiple of 8. | 5,120: 120 = 8 × 15 |
| 9 | The digits add up to a multiple of 9. | 3,708: 3 + 7 + 0 + 8 = 18 |
| 10 | The last digit is 0. | 6,430 ends in 0. |
| 11 | Add and subtract the digits in turn. The answer is 0 or a multiple of 11. | 2,728: 2 − 7 + 2 − 8 = −11 |
| 12 | The number passes the tests for 3 and for 4. | 1,512: 1 + 5 + 1 + 2 = 9, 12 = 4 × 3 |
The rules worth learning first
The rules for 2, 5 and 10 only look at the last digit, and most children see them on a hundreds chart before anyone names them. The rules for 3 and 9 add up the digits, and they are the ones that save the most time when factoring or simplifying a fraction. Learn those five first. The others are built from them, like 6, which is 2 and 3 together, and 12, which is 3 and 4.
The rule for 7 is slow, and many teachers skip it. Dividing by 7 is often just as quick. It is on the chart because children ask, and because it works for any number: do it again on the answer if the answer is still too big to know.
Where the rules get used
Fourth grade finds the factor pairs of a number to 100 and decides whether it is prime or composite (CCSS 4.OA.B.4), and the rules make that quick. Sixth grade uses them again for greatest common factors and least common multiples (6.NS.B.4). Keep the chart beside a factor sheet and ask the child to test each number before dividing.
Common questions
- Why does the digit sum work for 3 and 9?
- Every ten is 9 plus 1 and every hundred is 99 plus 1, so a number is always a multiple of 9 plus the sum of its digits. If the digit sum divides by 3 or 9, the whole number does too.
- Is a number divisible by 6 if it divides by 2 and 3?
- Yes. 2 and 3 share no factors, so passing both tests means it divides by 2 × 3, which is 6. The same is true of 3 and 4 for 12, but not of 2 and 4 for 8: 12 passes both and is not a multiple of 8.