Guide to Rules of Divisibility

Students of all levels including competitive examinations have to deal with figuring out if a number is divisible by another number or not. Doing the actual division to check the divisibility can be quite complicated and tiresome. Hence the students need to know the divisibility rules for such numbers. In the post below and subsequent posts (links given in this post) I’ve tried to give you a strong understanding of divisibility rules for various commonly required numbers. If you copy the below mentioned divisibility rules in a word doc and take a print out, it can act as a divisibility rules worksheet.

What are the divisibility rules?

Here we will take up one number at a time. First the rule of divisibility will be given, which will be followed by an example to illustrate the rule.

Divisibility by 2: If its unit digit is any of 0,2,4,6,8.
Ex : 100 is divisible by 2 while 101 is not.

Divisibility by 3: If the sum of its digits is divisible by 3.
Ex: 309 is divisible by 3, since sum of its digits = (3+0+9) = 12 , which is divisible by 3.

Divisibility by 4: If the number formed by the last two digits is divisible by 4

Ex: 2648 is divisible by 4, since the number formed by the last two digits is 48 which is divisible by 4.

Divisibility by 5: If its units digit is either 0 or 5.
Ex: 20825 and 50545 are divisible by 5.

Divisibility by 6: If it is divisible by both 2 & 3.
Ex: 53256 is divisible by 6 because it is divisible by 2 as well as 3.

Divisibility by 7: If after subtraction of a number consisting of the last three digits from a number consisting of the rest of its digits the result is a number that can be divided by 7 evenly

Ex.: 414141 is divisible 7 as 414-141= 273 is divisible by 7

Many different ways to test divisibility by seven have been devised. Some are long and complex, a few involve rewriting the digits, and one even consists of a grid-like box. We have chosen one of the more simplistic versions even though in almost every case it is quicker to merely perform long division.

Divisibility by 8: If the last three digits of the number are divisible by 8.
Ex: 3652736 is divisible by 8 because last three digits (736) is divisible by 8.

Divisibility by 9: If the sum of its digit is divisible by 9.
Ex: 672381 is divisible by 9, since sum of digits = (6+7+2+3+8+1) = 27 is divisible by 9.

Divisibility by 10: If the digit at unit place is 0 it is divisible by10.
Ex: 69410, 10840 is divisible by 10.

Divisibility by 11: If the difference of the sum of its digits at odd places and sum of its digits at even places, is either 0 or a number divisible by 11.
Ex: 4832718 is divisible by 11, since:
(Sum of digits at odd places) – (sum of digits at even places)
= (8+7+3+4)-(1+2+8) = 11

Divisibility by 12: A number is divisible by 12 if it is divisible by both 4 and 3.
Ex: 34632
(i) The number formed by last two digits is 32, which is divisible by 4
(ii) Sum of digits = (3+4+6+2) = 18, which is divisible by 3.

Divisibility by 13: Remove the last digit of a number. Multiply it by 4 and add it to the remaining truncated number.  Continue doing these steps until you reach a 2 digit number. If the result is divisible by 13, then so was the first number.

Example: 113945–>11394+20=11414–>1141+16=1157–>115+28=143 (since this number is divisible by 13, you can say 113945 is also divisible by 13)

You can go a step forward

14 + 12 = 26 is 2*13, so 113945 is divisible by 13.

Divisibility by 14: If a number is divisible by 2 as well as 7.

Divisibility by 15: If a number is divisible by both 3 & 5.

Divisibility by 16: If the number formed by the last 4 digits is divisible by 16.
Ex: 7957536 is divisible by 16, since the number formed by the last four digits is 7536, which is divisible by 16.

Divisibility by 24: If a number is divisible by both 3 & 8.

Divisibility by 40: If it is divisible by both 5 & 8.

Divisibility by 80: If a number is divisible by both 5 & 16.

I’ve seen people having trouble with finding the divisibility by little complicated numbers like 7, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47. In another post I’ve explained the divisibility rules for such complicated numbers

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Vineet Patawari

Hi, I'm Vineet Patawari. I fell in love with numbers after being scared of them for quite some time. Now, I'm here to make you feel comfortable with numbers and help you get rid of Math Phobia!

16 thoughts to “Guide to Rules of Divisibility”

  1. Mr. Vineet Patawari, great effort putting together all these rules. But, there is a problem with 7’s divisibility rule. It doesn’t work for odd digit numbers. Can you suggest a different method.

  2. I created a rule for divisibility by seven, eleven and thirteen whose algorithm for divisibility by seven is this:
    N = a,bcd; a’ ≣ ( − cd mod 7 + a ) mod 7; cd is eliminated and if 7|a’b then 7|N. The procedure is applied from right to left repetitively till the leftmost pair of digits is reached. If the leftmost pair is incomplete consider a = 0.
    Example: N = 382,536, using simple language:
    36 to 42 = 6; 6 + 2 − 7 = 1 → 15; 15 to 21 = 6; 6 + 3 − 7 = 2 → 28; 7|28 and 7|N.
    This rule is mentioned in my unpublished (officially registered) book: Divisibility by 7, the end of a myth?.

  3. Thanks Sir for your contribution.
    The divisibility of 7 can also be devised in this way, which I find bit easy and less time consuming. “Double the last digit of the given number and then subtract it from the number formed by the remaining digits and carry on this process”

    It’s always good to state by an example:

    Check whether 7923 is divisible by 7??

    Sol: 792-(2*3) = 786
    78-(2*6)= 66
    and 66 is not divisible by 7 thus 7923 is not evenly divisible by 7

    And for other bigger numbers for checking their divisibility, we can very well use the concept of co-prime thus reducing the calculation.

    Do post such informative stuff.

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