When a whole number is divisible by another number, the remainder being zero, then the second number is called the factor of the first number. Divisibility is a method to determine whether a number is completely divisible by the other or not. There are some easy tricks that can be used to determine if a number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, and 10, 11, 12, 13 (1 is easy as all numbers are divisible by 1).

Divisibility by 2: If a number is even or a number whose last digit (digit in the unit place) is an even number i.e.  2,4,6,8,10,12,14,16, including 0, it is always completely divisible by 2.

Example: 518 is an even number and is divisible by 2. The procedure to check whether 518 is divisible by 2 or not is as follows:

• Consider the number 518
• Just take the last digit 8 (unit digit place) and divide it by 2
• If the last digit 8 is divisible by 2 then the number 518 is also divisible by 2.

Divisibility by 3: A number is completely divisible by 3 if the sum of its digits is divisible by 3.

Example: Consider the number, 318. To check whether 318 is divisible by 3 or not use the following steps:

• Take the sum of the digits (i.e. 3+1+8= 12). 12 is a multiple of 3.
• Since 12 is divisible by 3, so the number 318 is also divisible by 3.

Divisibility by 4: Check the last two digits of the number. If the last two digits are divisible by 4, the number is divisible by 4.

Example: Take the number 4532.

• The Last two digits are; 32. 
• Since 32 is divisible by 4, hence the number 4532 is also divisible by 4.

Divisibility by 5: All the Numbers, ending with digits, 0 or 5 are always divisible by 5.

Example: 10, 10000, 100005, 795, 394554950, etc.

Divisibility by 6: Numbers which are divisible by both 2 and 3 are divisible by 6. That is, if the last digit of the given number is even and the sum of its digits is a multiple of 3, then the given number is also a multiple of 6.

Example: Take the number 648

• The digit is 8 which is divisible by 2.
• The sum of digits is 6+4+8 = 18, which is also divisible by 3. Hence, the number 648 is divisible by 6.

Divisibility by 7: The rule for divisibility by 7 is a bit complicated which can be understood by the steps given below:

• Multiply the last digit by 2
•  Subtract the product from the rest of the number.
•  If the result is divisible by 7, the number is divisible by 7.

Example: Take the number 343.

• Multiply the last digit by 2, 3 x 2 = 6
• Subtract the product from the rest of the number, 34 – 6 = 28
• Since 28 is divisible by 7, so the number 343 is also divisible by 7.

Divisibility by 8: If the last three digits of a number are divisible by 8, then the number is completely divisible by 8.

Example: Talke the number 3168.

• The last 3 digit is 168 and is divisible by 8.
• Therefore the number 3168 is also divisible by 8 (396 x 8=3168).

Divisibility by 9: Add all the digits in the number. If the sum is divisible by 9, the number is also divisible by 9.

Example: take the number 342.

• Add the digits of the number 3+4+2 = 9, which is divisible by 9.
• Therefore the number 342 is also divisible by 9.

Divisibility by 10: If the last digit of the number is 0 the number is divisible by 10.

Example: 10, 20, 30, 40, 100, 23450, 23586352360, and so on

Divisibility by 11: If the difference of the sum of alternative digits of a number is divisible by 11, then that number is divisible by 11 completely. In order to check whether a number like 1826 is divisible by 11, use the following steps.

• Group the alternative digits i.e. digits which are in odd places together and digits in even places together. Here 12 and 86 are two groups.
• Take the sum of the digits of each group i.e. 1+2=3 and 8+6= 14
• Now find the difference of the sums; 14-3=11
• If the difference is divisible by 11, then the original number is also divisible by 11. Here 11 is the difference which is divisible by 11.
• Therefore, 1826 (166 x 11) is divisible by 11.

Divisibility by 12: If the number is divisible by both 3 and 4, then the number is divisible by 12 exactly. 

Example: Take the number 5864.

• Sum of the digits = 5 + 8 + 6 + 4 = 23 (not a multiple of 3)
• Last two digits = 64 (divisible by 4)
• The given number 5864 is divisible by 4 but not by 3; hence, it is not divisible by 12.

Divisibility by 13: For any given number, to check if it is divisible by 13, we have to add four times of the last digit of the number to the remaining number and repeat the process until we get a two-digit number.  Now check if that two-digit number is divisible by 13 or not. If it is divisible, then the given number is divisible by 13.

Example: Take the number 2795 

• 4 times of the last digit 5 is (5 x 4), add this to the remaining number 279 which is equal to  279 + (5 x 4)= 279 + 20 =299
• Continue the same process outline in the 1st step. 29 + (9 x 4) = 29 + 36 = 65.
• The number 65 is divisible by 13, 13 x 5 = 65. Hence the number 2795 is divisible by 13 (215 x 13 = 2795).

Divisibility Rules – Important Points to Remember

• If a number is divisible by another number, then it is divisible by each of the factors of that number.
• If a number is divisible by two co-prime numbers, then it is divisible by their product also.
• If two given numbers are divisible by a number, then their sum is also divisible by that number.
• If two given numbers are divisible by a number, then their difference is also divisible by that number.