Divisibility by different numbers

Divisibility by 2, 4, 8, 16

• A number is divisible by 2 if the last digit of the number is divisible by 2 i.e. last digit is 0, 2, 4, 6 or 8

• A number is divisible by 4 if the last two digits of the number are divisible by 4. For eg. 1348 is divisible by 4 because 48 is divisible by 4.

• A number is divisible by 8 if the last three digits of the number are divisible by 8. For eg. 21348 is not divisible by 8 because 348 is not divisible by 8.

• A number is divisible by 16 if the last four digits of the number are divisible by 16. For eg. 1,134,224 is divisible by 16 because 4224 is divisible by 16.

Divisibility by 3, 9, 27

• A number is divisible by 3 if the sum of all digits of the number are divisible by 3. Hence 729 (sum of digits = 7+2+9=18) is divisible by 3

• A number is divisible by 9 if the sum of all digits of the number are divisible by 9. Hence 62172 (sum of digits = 6+2+1+7+2=18) is divisible by 9

• A number is divisible by 27 if the sum of blocks of 3 from left to right are divisible by 27. Hence 134433 (sum of blocks = 134+433=567) is divisible by 2

Divisibility by 5, 25, 125,625

• A number is divisible by 5 if the last digit of the number is 0 or divisible by 5.
• A number is divisible by 25 if the last two digits of the number is 00 or divisible by 25.
• A number is divisible by 125 if the last three digits of the number is 000 or divisible by 125.
• A number is divisible by 625 if the last four digits of the number is 0000 or divisible by 625.

Divisibility by 6: Divisible by both 2 and 3 

Divisibility by 12: Divisible by both 3 and 4

Divisibility by 7

• Remove the last digit, double it and subtract it from the truncated original number. For example: 1638, 163-16=147 is divisible by 7 hence 1638 is divisible by 7. This process should be repeated for larger numbers.

Divisibility by 11

• Take sum of all even digits and subtract it from the sum of all odd digits. If the difference is a multiple of 11 the original number will be a multiple of 11. Example 23452 (sum odd = 2+4+2=8 sum even=3+5=8) difference=8-8=0 is divisible by 11 hence 23452 is divisible by 11.

Divisibility properties

• For composite divisors, check if the number is divisible by the factors individually. Hence to check if a number is divisible by 6 it must be divisible by 2 and 3.

Quick Recap

• Divisibility by 2: Last digit divisible by 2

• Divisibility by 4: Last two digits divisible by 4

• Divisibility by 8: Last three digits divisible by 8

• Divisibility by 16: Last four digit divisible by 16

• Divisibility by 3: Sum of digits divisible by 3

• Divisibility by 9: Sum of digits divisible by 9

• Divisibility by 27: Sum of blocks of 3 (taken right to left) divisible by 27

• Divisibility by 7: Remove the last digit, double it and subtract it from the truncated original number. Check if number is divisible by 7

• Divisibility by 11: (sum of odd digits) - (sum of even digits) should be 0 or divisible by 11