DIVISIBILITY RULES

The DIVISABILITY rule is a concise and useful way to determine whether a given integer is divisible by a fixed divisor, usually by examining its digits without performing the Division.

Let us see the divisibility rules of various numbers:

• There is no specific condition for divisibility by '1', Any integer is divisible by '1'.
• A number is divisible by ‘2’, when its unit digit is either  ‘Zero’ Or ‘Even number’ (2,4,6, or 8).

Number	Last Digit	Even or Odd	Is divisible by 2 as per rule?	Remainder after actual calculation
124	4	Even	Yes	0
67485	5	Odd	No	1
1245987	7	Odd	No	1

• A number is divisible by ‘3’, when the sum of its digits is divisible by ‘3’.

Number	Sum of the Digits	Is divisible by 3 as per rule?	Remainder after actual calculation
124	1+2+4 = 7	No	1
67485	6+7+4+8+5 = 30	Yes	0
1245987	36	Yes	0

• A number is divisible by ‘4’, when the number formed by the two extreme right end digits is either divisible by '4' or both these digits are zero’s.

Number	Number formed by last two digits	Number formed by last two digits divisible by 4	Number divisible as per rule?	Remainder after actual calculation
45	45	No	No	1
124	24	Yes	Yes	0
67485	85	No	No	1
1245987	87	No	No	3

• A number is divisible by ‘5’, when its unit digit is either 'zero' or '5'.

Number	Last Digit	Number divisible by 5 as per rule?	Remainder after actual calculation
45	5	Yes	0
124	4	No	4
67480	0	Yes	0

• A number is divisible by ‘6’, when it is divisible bye '2' as well as '3'.

Number	Number Divisible by 2?	Number Divisible by 3?	Number divisible by 6 as per rule?	Remainder after actual calculation
45	No	Yes	No	3
124	Yes	No	No	4
67482	Yes	Yes	Yes	0

Note: In the above examples if the number is not divisible any one of 2 & 3 or both the numbers then the number is not divisible by 6.

• A number is divisible is by 7, 

 

• A number is divisible by ‘8’, when the number formed by its three extreme right end digits is divisible by 8 or when these last three digits are Zero’s.

Number	Number formed by last three digits	Number formed by last two digits divisible by 8?	Number divisible as per rule?	Remainder after actual calculation
445	445	No	No	5
128	128	Yes	Yes	0
67485	485	No	No	5
1245987	987	No	No	3

• A number is divisible by ‘9’, when the sum of its digits is divisible by 9.

Number	Sum of the Digits	Is divisible by 9 as per rule?	Remainder after actual calculation
124	1+2+4 = 7	No	1
67485	6+7+4+8+5 = 30	No	3
1245987	36	Yes	0

• A number is divisible by ‘10’, when its unit digit is zero.

Number	Last Digit	Number divisible by 10 as per rule?	Remainder after actual calculation
45	5	No	5
124	4	No	4
67480	0	Yes	0

Note: If the last digit of the number is zero then the number is divisible by 10 

• A number is divisible by ‘11’, when the difference between the sums of the alternate digits is either zero or divisible by 11.

Number	Sum of the digits in Odd Positions	Sum of the digits in Even Positions	Difference between the sums	Number divisible by 11 as per rule?	Remainder after actual calculation
45	5	4	5-4 = 1	No	1
124	4+1 = 5	2	5-2 = 3	No	3
67480	0+4+6 = 10	8+7 = 15	15-10 = 5	No	5
96419818	1+9+4+9 = 23	8+8+1+6 = 23	23-23 = 0	Yes	0

• A number is divisible by ‘12’, when it is divisible by ‘3’ as well as ‘4’.
• A number is divisible by ‘13’, if sum of 4 times the digit in units place and the number in the remaining part is multiple of 13.
• If the difference of 5 times the digit in units place and the number in the remaining part is '0' or multiple of 17, then the number is divisible by 17
• If the sum of double the digit in units place of a given number and number in the remaining part is multiple of 19, then the given number is divisible of 19.
• A number is divisible by ‘25’, when its last two digits are  "00, 25, 50 or 75".
• A number is divisible by ‘50’, when its last two digits are  "00, 50".
• A number is divisible by ‘75’, when its last two digits are  "00, 25, 50 or 75", and the sum of all the digits must be divisible by 3.