Divisibility Rule:

A divisibility test is an easy way to identify whether the given number is divided by a fixed divisor without actually performing the division process. 

“If a number is completely divided by another number, then the quotient should be a whole number and the remainder should be zero.”

Test of Divisibility by 3: 

A number is divisible by 3 if the sum of the digits is divisible by 3.

Examples:

i. Take the number 814728. Add the digits 8 + 1 + 4 + 7 + 2 + 8 = 30. 30 is divisible by 3. Hence 814728 is divisible by 3.
ii. Take another number 349709. Add the digits 3 + 4 + 9 + 7 + 0 + 9 = 32. 32 is not completely divisible by 3. Hence 349709 is not divisible by 3.

Test of Divisibility by 4:

A number is divisible by 4 is the number formed by the last two digits is divisible by 4.

Examples:

i. Take a number 23424224. The last two digits form the number 24. Hence the number is divisible by 4.
ii. Take the number 234242241. The last two digits form the number 41. Hence the number is not divisible by 4.

Test of Divisibility by 5:

A number is divisible by 5 if the unit digit is either 0 or 5.

Examples:

i. Take the number 234242245. The last digit is 5. Hence the number is divisible by 5.
ii. Take the number 2342422451. The last digit is 1. Hence the number is not divisible by 5.

Test of Divisibility by 6:

A number is divisible by 6 if it is divisible both by 2 and 3.

Example:

i. Take the number 753222. The last digit is 2, hence divisible by 2. The sum of the digits is 21. 21 is divisible by 3. Hence the number is divisible both by 2 and 3.
Hence the number is divisible by 6.

Test of Divisibility by 8:

A number is divisible by 8, if the number formed by the last 3 digits is divisible by 8.

Examples:

i. Take a number 293512. The number formed by the last three digits is 512. 512 is
divisible by 8. Hence the number 293512 is divisible by 8.
ii. Take a number 293513. The number formed by the last three digits is 513. 513 is
not divisible by 8. Hence the number 293512 is divisible by 8.

Test of Divisibility by 9:

A number is divisible by 9 if the sum of the digits is divisible by 9.

Examples:

i. Take a number 874728. Add the digits 8 + 7 + 4 + 7 + 2 + 8 = 36. 36 is divisible by 9. Hence 874728 is divisible by 9.
ii. Take another number 349709. Add the digits 3 + 4 + 9 + 7 + 0 + 9 = 32. 32 is not completely divisible by 9. Hence 349709 is not divisible by 9.

Test of Divisibility by 10:

A number is divisible by 10, if its unit digit is 0.

Examples:

10, 20, 3430, 23249860 are all divisible by 0.

Test of Divisibility by 11:

A number is divisible by 11, if the difference between the sum of its digits at odd places and the sum of the digits at even places is either 0 or a number divisible by 11.

Examples:

i. Take a number 8192657. The sum of the digits at odd places = (7 + 6 + 9 + 8) = 30. The sum of the digits at even places = (5 + 2 + 1) = 8. The difference is 22 which is divisible by 11. Hence the number 8192657 is divisible by 11.
ii. Take a number 8192656. The sum of the digits at odd places = (7 + 6 + 9 + 6) = 29. The sum of the digits at even places = (5 + 2 + 1) = 8. The difference is 21 which is not divisible by 11. Hence the number 8192657 is not divisible by 11.

More Examples:

14 is divisible by 7, because 14 ÷ 7 = 2 exactly

15 is not divisible by 7, because 15 ÷ 7 = 2 17 (the result is not a whole number)

0 is divisible by 7, because 0 ÷ 7 = 0 exactly (0 is a whole number)

“Divisible by” and “can be exactly divided by” mean the same thing.