Divisibility Rules for 13: Method, Solved Questions

The term ‘divisibility’ is used to check whether the number is totally divisible by another number or not, and leaves 0 as the remainder. Through the rules of divisibility, we can find out whether the number is fully divisible or not without performing any division operation. The rule of divisibility of 13 helps us to check if a number is divisible by 13 or not. There are divisibility rules of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,13,14, etc. For example, if we have a number, say 22, we can simply say that 22 is divisible by 2, because it has an even number in its unit digit. If it would have an odd number in its unit place, then it would not be divisible by 2. So from the rules of divisibility, we can check the divisibility of the numbers very easily.

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Key takeaways: divisibility, divisibility rule of 13, divisibility rule of 14, divisibility rule of 17, rules of divisibility, how to check divisibility.

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What is the divisibility rule of 13?

The divisibility rule of 13, is a set of rules through which, we can check whether a particular number is completely divisible by 13 or not.

A number to be divisible by 13, must have one digit which is a multiple of 4, and the product of the digits when added to the rest of the number, must give 0 or a multiple of 13. That can be said in another way, that a number is divisible by 13 when the sum obtained after adding the product of multiplying the units digit by 4 and the rest of the number to its left should be 0 or should not leave any remainder other than 0 upon division by 13. There are 4 rules of divisibility by 13.

Check Important Notes for Euclid’s Division Lemma

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Rules of Divisibility

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Rule 1:

The first rule of divisibility by 13, needs the number to be grouped in a set of 3, starting from the right. Now, the operations of addition and subtraction need to be applied alternatively from the rightmost group of 3 digits and then find the result. If one gets the result to be 0 or a number that can be divided by 13, that would mean that the number is totally divisible by 13.

For example:

To check if the number 2,678,450 is divisible by 13 or not, we apply rule 1. As the rule states applying the subtraction and addition operations alternatively from the rightmost group of 3 digits, we get 450+678-2 =1128-2= 1126. When 1126 is not divisible by 13, it leaves a reminder. Thus 1,139,502 is exactly divisible by 13.

Rule 2:

The second rule of divisibility is that the digit in the unit place must be multiplied by 4, and then the product must be added to the rest of the number. Thus the number obtained must either be 0 or the multiples of 13, then the number is divisible by 13.

For example:

In the number 832, is divisible by 13 or not, we apply rule 2. It states that the one place digit is 2, and by Multiplying the one place digit by 4, we get (4 x 2), which is 8. Now adding 8 to the rest of the digits, we get 83 + 8 = 91. As 91 is a multiple of 13, therefore 832 is divisible by 13.

Rule 3:

The third rule of checking the divisibility by 13 is that the last two digits of the number must be taken and then subtracted from the product of 4 and the rest of the numbers. The resulting number must be 0 or divisible by 13 in order for the number to be divisible by 13.

For example:

To check if the number 745, the last two digits are 45. 4 is multiplied by the rest of the digits, that is 7, we get 28. Now 28 is subtracted from 45(the last two digits of the number), we get 17. Therefore, 17 is not a multiple of 13, thus 745 is not divisible by 13. This method of divisibility by 13 can be easily applied and very effective with three-digit numbers.

Rule 4:

The fourth rule states that the digit in the unit place must be multiplied by 9, and the difference between the product and the rest of the numbers needs to be found. If the resulting number obtained is 0 or a number that is multiple by 13, then the given number is divisible b 13.

For example:

Following the fourth rule of checking the divisibility of the number 987, multiplying the last digit (7) with 9 we get 7 x 9, which is 63. Now 63 is subtracted from 79, and we get 16. Since 16 is not a multiple of 13, we can say that 987 is not divisible by 13.

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Divisibility Rule of 13 and 14

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The rules of divisibility rules of 13 and 14 are different. For the divisibility of 13, the above 4 rules must be maintained. And for the divisibility of 14, the number must be divisible by both 2 and 7.

For example:

156 is divisible by 13, as the last digit is multiplied by 4, and it comes to 4*4= 24. Now when 24 is added to 15 (the rest of the numbers), it comes to 39. And 39 is divisible by 13. Thus 156 is divisible by 13.

156 is not divisible by 14, because even if 156 is divisible by 2, it is not divisible with 7. Thus 156 is not divisible with 14.

Divisibility test of 13 and 17.

The process of checking the divisibility rule of 13 and 17 is different. For checking the divisibility of 13, the above rules can be followed. The divisibility by 17 can be checked by multiplying the unit digit by 5 and then subtracting the product from the rest. Now if the result is a multiple of 17 or 0, then the number is divisible by 17.

For example:

187 is not divisible by 13, because, 7*4= 28. Then the product is added to 18, thus the result comes to 46. 46 is not a multiple of 13, thus it is not divisible by 13.

187 is total divisible by 17, because, 5*7=35, now 17 is subtracted from 35, we get 17, which is a multiple of 17. Thus 187 is totally divisible by 17.

Also Check:

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Things to remember

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• There are four rules to check the divisibility by 13.
• The first rule is that the given number is grouped by taking 3 digits at a time from the one’s place. Then performing addition and subtraction alternatively, we will come to a result which must be 0 or a multiple of 13, for the numbers to be exactly divisible by 13.
•  The second rule is that the digit in the unit place is multiplied by 4, and then the product is added to the rest of the number, and the results must be 0 or a multiple of 13, for the numbers to be exactly divisible by 13.
• The third rule is that, by taking the last two digits, multiplying 4 with the rest of the digits, and then subtracting the product from the two digits, the resultant must be 0 or a multiple of 13, for the numbers to be exactly divisible by 13.
• The fourth rule is by multiplying the unit digit with 9 and then finding the difference between the product and the rest of the numbers. The resultant must be 0 or a multiple of 13, for the numbers to be exactly divisible by 13.

Check Also: Class 6 Mathematics Chapter 14 Practical Geometry

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