Arithmetic & Numbers · Grades 4–8
Divisibility Rules for 2 to 10, Explained
Rules by last digits
- 2: the last digit is even
- 5: the last digit is 0 or 5
- 10: the last digit is 0
- 4: the last two digits are divisible by 4
- 8: the last three digits are divisible by 8
Rules by digit sum
A number is divisible by 3 when its digits add to a multiple of 3, and by 9 when they add to a multiple of 9. This works because every power of ten leaves a remainder of 1 when divided by 3 or 9.
1,236
- Digit sum 12
Step-by-step working 2 steps
- 1Apply each divisibility rule
÷ 2: yes (last digit 6 is even) ÷ 3: yes (digit sum 12) ÷ 4: yes (last two digits 36) ÷ 5: no (last digit 6) ÷ 6: yes (must be divisible by both 2 and 3) ÷ 7: no (remainder when divided by 7 is 4) ÷ 8: no (last three digits 236) ÷ 9: no (digit sum 12) ÷ 10: no (last digit 6)
Rules: 2, 5, 10 look at the last digit; 4 and 8 at the last two or three digits; 3 and 9 at the digit sum; 6 needs 2 and 3.
- 2Result
1236 is divisible by 2, 3, 4, 6.
Combining rules
A number is divisible by 6 if it passes both the 2-rule and the 3-rule. For 360 the result is Divisible by 2, 3, 4, 5, 6, 8, 9, 10.
360
- Digit sum 9
Step-by-step working 2 steps
- 1Apply each divisibility rule
÷ 2: yes (last digit 0 is even) ÷ 3: yes (digit sum 9) ÷ 4: yes (last two digits 60) ÷ 5: yes (last digit 0) ÷ 6: yes (must be divisible by both 2 and 3) ÷ 7: no (remainder when divided by 7 is 3) ÷ 8: yes (last three digits 360) ÷ 9: yes (digit sum 9) ÷ 10: yes (last digit 0)
Rules: 2, 5, 10 look at the last digit; 4 and 8 at the last two or three digits; 3 and 9 at the digit sum; 6 needs 2 and 3.
- 2Result
360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10.
The awkward one: 7
There is a quick trick for 7, but the remainder test is more reliable. For 1001 the only divisor from 2 to 10 is: Divisible by 7.
Common mistakes
Using the digit sum for 4 or 8
These depend on the last two or three digits, not the sum.
Forgetting both conditions for 6
Divisible by 2 and by 3 are both required.
Mis-adding digits
Re-add the digit sum if it is still large.
Frequently asked questions
Is there a rule for 11?
Yes: the alternating sum of the digits must be a multiple of 11.
