"Divisible By" means "when you divide one number by another the result is a whole number"
Divisibility test is a quick way of finding out whether a number is divisible by another number or not. Is 8 divisible by 2 387?
It would difficult to know this without carrying out the long division.
Note: These are the tests we learnt in various classes notably; divisibility tests of 2, 3, 4, 5, 6, 8, 9 and 11
Example 1
Which of the following numbers are divisible by 2? (a) 230 (c) 598 (b) 209 (d) 947
Solution
(a) 230 is divisible by 2 because the last digit is 0.
(b) 209 is not divisible by 2 because the last digit is 9.
(c) 598 is divisible by 2 because the last digit is 8.
(d) 947 is not divisible by 2 because the last digit is 7.
A number is divisible by 4 if the last two digits form a number that is divisible by 4.
Example 1
Which of the following numbers is divisible by 4?
(a) 219 (b) 4 441 (c) 4 872 (d) 430
Solution
(a) The last two digits in 219 are 19. 19 is not divisible by 4, therefore, 219 in not divisible by 4.
(b) The last two digits in 4 441 are 41. 41 is not divisible by 4, therefore, 4 441 is not divisible by 4.
(c) The last two digits in 4 872 are 72. 72 is divisible by 4 therefore, 4 872 is divisible by 4.
(d) The last two digits in 430 are 30. 30 is not divisible of 4 hence 430 is not divisible by 4.
The correct answer is (c) 4 872.
A number is divisible by 8 if the last three digits form a number which is divisible by 8.
Which of the following numbers is divisible by 8?
(a) 4 936 (b) 12 975 (c) 9 243 (d) 14 725
Solution
(a) The last three digits in 4 936 are 936. 936 is divisible by 8 (936 ÷ 8 = 117). 4 936 is divisible by 8.
(b) The last three digits in 12 975 are 975. 975 is not divisible by 8. 12 975 is not divisible by 8.
(c) The last three digits in 9 243 are 243. 234 is not divisible by 8. 9 243 is not divisible by 8.
(d) The last three digits in 14725 are 725. 725 is not divisible by 8. 14 725 is not divisible by 8.
The correct answer is therefore (a) 4 936.
A number is divisible by 3 if the sum of its digits is divisible by 3.
Example 1
Which of the following numbers is divisible by 3?
(a) 1 111 (b) 9 585 (c)206 (d) 91 403
Solution
(a) The sum of the digits of 1 111 is 1 + 1 + 1 + 1 = 4. 4 is not divisible by 3. 1 111 is not divisible by 3.
(b) The sum of the digits of 9 585 is 9 + 5 + 8 + 5 = 27. 27 is divisible by 3. 9585 is divisible by 3. (Note that you can also get the sum of the digits of 27 i.e. 2 + 7 to get 9. 9 is divisible by 3. The number 9 585 is divisible by 3).
(c) The sum of the digits of 206 is 2 + 0 + 6 = 8. 8 is not divisible by 3 206 is not divisible by 3.
(d) The sum of the digits of 91 403 is 9 + 1 + 4 + 0 + 3 = 17. 17 is not divisible by 3 and so 91 403 is not divisible by 3.
The correct answer is (b) 9 585.
A number is divisible by 6 if it is also divisible by 2 and 3 i.e. if its last digit is 0, 2, 4, 6, 8 and the sum of its digits is divisible by 3.
Example 1
Which of the following numbers is divisible by 6?
(a)106 (b) 4 981 (c)1 138 (d) 2 166
Solution
(a) The sum of digits in 106 is 1 + 0 + 6 = 7. 7 is not divisible by 3. 106 is not divisible by 6.
(b) The last digit in 4 981 is 1 thus its not divisible by 2. The sum of digits in 4 981 is 4 + 9 + 8 + 1 = 22. 22 is not divisible by 3. 4 981 is not divisible by 6.
(c) The sum of digits in 1 138 is 1 + 1 + 3 + 8 = 13. 13 is not divisible by 3. 1 138 is not divisible by 6.
(d) The sum of digits in 2 166 is 2 + 1 + 6 + 6 = 15. 15 is divisible by 30. Therefore, 2 166 is divisible by 6.
The correct answer is (d) 2 166.
Just like the divisibility test for 3.
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example 1
Which of the following numbers is divisible by 9?
(a) 7 324 (b) 18 948 (c) 19 327 (d) 48735
Solution
(a) The sum of the digits in 7 324 is 7 + 3 + 2 + 4 = 16. It is not divisible by 9. 7 324 is also not divisible by 9.
(b) The sum of the digits is 18 948 is 1 + 8 + 9 + 4 + 8 = 30. 30 is not divisible by 9. 18 948 is also not divisible by 9.
(c) The sum of the digits in 19 327 is 1 + 9 + 3 + 2 + 7 = 22. 22 is not divisible by 9. 19 327 is also not divisible by 9.
(d) The sum of the digits in 48 735 is 4 + 8 + 7 + 3 + 5 = 27. 27 is divisible by 9. 48 735 is divisible by 9.
The correct answer is therefore (d) 48 735.
5 A number is divisible by 5 if the last digit ends with a zero (0) or five (5).
Example 1
Which of the following numbers are divisible by 5?
(a) 154 (b) 740 (c) 3 965 (d) 1 783
Solution
(a) 154 is not divisible by 5 because the last digit is 4.
(b) 740 is divisible by 5 because the last digit is 0.
(c) 3 965 is divisible by 5 because the last digit is 5.
(d) 1 783 is not divisible by 5 because the last digit is 3.
The correct answers are (b) 740 and (c) 3 965.
A number is divisible by 10 if the last digit is zero (0).
Note: All numbers divisible by 10 are also divisible by 2 and also by 5.
Example 1
Which of the following numbers is divisible by 10.
(a) 189 (b) 4 340 (c) 9 356 (d) 1 074
Solution
189, 9 356 and 1 074 are not divisible by 10 because the last digits are 9, 6, 4 respectively.
4 340 is divisible by 10 because the last digit is 0.
The correct answer is (b) 4 340.
A number is divisible by 11 if the difference of the sum of the alternative digits is 0, 11, or a multiple of 11.
Example 1
Which of the following numbers is divisible by 11?
(a) 29 342 (b) 29 301 (c) 41 349 (d) 11 111
Solution
(a) Get the sum of the alternate digits in 29 342. 2 + 3 + 2 = 7 9 + 4 = 13 Then find their difference 13 – 7 = 6. 6 is not divisible by 11. Therefore, 29 342 is not divisible by 11.
(b) Get the sum of the alternate digits in 29 301. 9 + 0 = 9 2 + 3 + 1 = 6 Then find their difference 9 – 6 = 3. 3 is not divisible by 11. Therefore, 29 301 is not divisible by 11.
(c) Get the sum of the alternate digits in 41 349. 1 + 4 = 5 4 + 3 + 9 = 16 Then find their difference 16 – 5 = 11. 11 is divisible by 11 therefore, 41 349 is divisible by 11.
(d) Get the sum of the alternate digits in 11 111. 1 + 1 = 2 1 + 1 + 1 = 3 Then find their difference 3 – 2 = 1 1 is not divisible by 11 and therefore 11 111 is not divisible by 11.
Therefore the correct answer is (c) 41 349
A number is divisible by 11 if the sum of the paired digits is divisible by 11.
Example 2
Which of the following numbers is divisible by 11? (a) 150 161 (b) 79 218 (c) 620 182 (d) 76 379
Solution
(a) From right to left, pair the digits in 150 161 into 15, 01, 61 then add the pairs 15 + 01 + 61 = 77 77 is divisible by 11 therefore, 150 161 is divisible by 11.
(b) From right to left, pair the digits in 79218 into 7, 92,18 then add the pairs 18 + 92 + 07 = 117. Note: The digit 7 has been written as 07 to make it a pair. 117 is not divisible by 11, therefore 79218 is not divisible by 11.
Note:
We can still pair 117 as 1,17 then find the sum 17 + 01 = 18. 18 again is not divisible by 11. Therefore, 79 218 is not divisible by 11.
(c) From right to left, pair the digits in 620 182 into 62,01,82 then add them up 62 + 01 + 82 = 145. 145 is not divisible by 11 so, 620 182 is not divisible by 11 We can still pair 145 again as 1,45 and find the sum. 1 + 45 = 46. 46 is not divisible by 11 hence the whole number 620 182 is not divisible by 11.
(d) From right to left, pair the digits in 76 379 into 07,63,79 and find their sum 07 + 63 + 79 = 149. Pair the sum further 1,49 find their sum 1 + 49 = 50 50 is not divisible by 11 therefore, 76 379 is not divisible by 11.
The correct answer is (a) 150161.
Video on the why of the 3 divisibility rule
Example 3
Which of the numbers below is divisible by 11? (a) 438 120 (b) 321 387
Solution
(a) The sum of the alternate digits in 438 120 is 14 and 4, and their difference is 14 – 4 = 10. 10 is not divisible by 11 so 432 120 is not divisible by 11.
(b) The sum of the alternate digits in 321 387 is 12 and 12 and their difference is 12 – 12 = 0 therefore 321 387 is divisible by 11.
The correct answer is (b) 321 387.
watch divisibility rules video
This chart will help you:
This chart will help you:
