Divisibility Rule of 13
The divisibility rule of 13 states that a number is divisible by 13 when the ones place digit of a number is multiplied by 4 and the product is added to the rest of the number either gives 0 or a multiple of 13. In other words, 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.
What is the Divisibility Rule of 13?
Divisibility is the term used to check if a number completely divides another number with 0 as the remainder. Divisibility rules in math help us to easily find out if a number is divisible by another number without performing division operations. The most commonly used divisibility rules are from 2 to 13. The divisibility rule of 13 helps us to determine if a number is divided by 13 without leaving any remainder. There are 4 divisibility rules to check if a number is divisible by 13 or not. Let's study these 4 rules in detail:
Divisibility by 13 Rule 1:
Group the given number into sets of 3 starting from the right, or the unit place. From the rightmost group of 3 digits apply the subtraction and addition operations alternatively and find the result. If the result is either a 0 or it can be divided by 13 completely without leaving a remainder, then the number is divisible by 13.
For example, in the number 1,139,502 applying the subtraction and addition operations alternatively from the rightmost group of 3 digits, we get 502139+1 = 364. 364/13 gives 28 as quotient and 0 as remainder. Therefore, 1139502 is divisible by 13.
Divisibility by 13 Rule 2:
Multiply the unit place digit by 4, and add the product with the rest of the number to the left of the unit place digit. If the resulting number is a 0 or multiples of 13, then the number is divisible by 13.
For example, in the number 416, the ones place digit is 6. Multiplying the ones place digit by 4, we get (4 x 6), which is 24. Adding 24 to the rest of the digits to the left, we get 41 + 24 = 65. Since 65 is a multiple of 13, 416 is divisible by 13.
Divisibility by 13 Rule 3:
Take the last two digits of a number and subtract it from the product of 4 and the rest of the number. If the resulting number is 0 or a multiple of 13, then we can say that the number is divisible by 13. For example: In the number 520, the last two digits are 20. Product of 4 and the rest of the number (5) is 5 x 4, which is 20. On subtracting them, we get 2020, which is 0. Therefore, 520 is divisible by 13. This method of divisibility by 13 can be easily applied and very effective with threedigit numbers.
Divisibility by 13 Rule 4:
Multiply the digit at the unit place by 9 and find the difference between the product obtained and the rest of the number to its left. If the number is 0 or a multiple of 13, then we can say that the given number is divisible by 13.
For example, in the number 793, multiplying the last digit (3) with 9 we get 3 x 9, which is 27. On subtracting this from the rest of the number, which is 79, we get 7927, which is 52. Since 52 is a multiple of 13, we can say that 793 is divisible by 13.
Divisibility Rule of 13 for Large Numbers
Divisibility of a number by 13 is simple if the numbers are of 2 digits or if we are familiar with the first few multiples of 13. But what do we do when we are given a larger number and asked to check for its divisibility by 13? As discussed in the section above, there are 4 rules to find whether a number is divisible by 13 or not. Now let us take a 5digit number and apply one of the rules to check its divisibility by 13.
Divisibility Rule of 13 and 14
The divisibility rules of 13 and 14 are different. By the divisibility rule of 13, a number is said to be divisible by 13, if the product of 4 and the last digit of the number is added to the rest of the number results in a 0 or a multiple of 13. The divisibility rule of 14 states that for a number to be divisible by 14, it should be divisible by 2 and 7. Let us take an example and find if a number is divisible by 13 and 14.
Divisibility by 13 for 156  Divisibility by 14 for 156 

Multiply the last digit by 4. 4 × 6 = 24. Add it to the rest of the digits, which is 15. Therefore, 15 +24 = 39.  Is 156 divisible by 2? Yes, since 156/2 = 0. Is 156 divisible by 7? To check that, multiply the last digit by 2. It gives (6 × 2 = 12). Subtract it with the rest of the number (15). 1512 =3. 
Is 39 a multiple of 13? Yes. Hence 156 is divisible by 13.  Is 3 a multiple of 7? No, hence 156 is not divisible by 14. 
Divisibility Test of 13 and 17
As 13 and 17 are two different numbers, their divisibility rules are also different. We know that there are four ways of finding whether a number is divisible by 13. To check if a number is divisible by 17, we multiply the unit place digit of a number by 5 and subtract the product from the rest of the number. If the difference is a multiple of 17 or 0, then the given number is divisible by 17. Let us check the divisibility rules of 13 and 17 with the number 187.
Divisibility by 13 for 187  Divisibility by 17 for 187 

Multiply the last digit by 4. 4 × 7 = 28. Add it to the rest of the number, which is 18. Therefore, 18 +28 = 46.  Multiply the last digit by 5. 5 × 7 = 35. Find the difference between the product and the rest of the number, which is 18. Therefore, 35 18 = 17. 
Is 46 a multiple of 13? No, hence 187 is not divisible by 13.  Is 17 a multiple of 17? Yes, hence, 187 is divisible by 17. 
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Divisibility Rule of 13 Examples

Example 1: Using the divisibility rule of 13, find if 2769 is divisible by 13 or not.
Solution:
Follow the following steps to check if 2769 is divisible by 13 or not. Let us use the following rule to check this.
Multiply the last digit by 9 and find the difference between the product and the rest of the number to the left. If the resulting number is a multiple of 13, then the number is divisible by 13.Step 1: Multiply the last digit by 9. (9 × 9) = 81
Step 2: Subtract 81 from 276. 276  81 = 195. Is 195 a multiple of 13? We are not sure. Let us repeat the process again.
Step 3: Multiply the last digit of 195 by 9, which is 5 × 9 = 45.
Step 4: Find the difference between 45 and the rest of the number (19), which is 4519 = 26.
Step 5: Is 26 a multiple of 13? Yes, therefore, 2769 is divisible by 13. 
Example 2: Perform the divisibility test of 13 on 3,049,514 and find if it is divisible by 13 or not.
Solution:
Let us use the following rule to find if the number 3,049,514 is divisible by 13 or not.
Group the given number into sets of 3 starting from the right. From the rightmost group of 3 digits apply the subtraction and addition operations alternatively and find the result. If the result is either a 0 or it can be divided by 13 completely without leaving a remainder, then the number is divisible by 13.On grouping the number into sets of 3, and applying the subtraction and additions operations alternatively we get,
514  049 + 3 = 468.
When 468 is divided by 13, we get the quotient as 36 and the remainder as 0. Hence, 3,049,514 is divisible by 13. 
Example 3: Can you find if the smallest 4digit number is divisible by 13?
Solution:
The smallest 4digit number is 1000. Let us apply the following rule to find if 1000 is divisible by 13 or not.
Take the last two digits of a number and subtract it from the product of 4 and the rest of the number. If the resulting number is 0 or a multiple of 13, then we can say that the number is divisible by 13.
The last two digits of the number are 00. The product of 4 with the rest of the number (10) is (10 × 4), which is 40. Subtracting 0 from 40, we get 40  0, which is 40. But 40 is not a multiple of 13. Hence, the smallest 4digit number is not divisible by 13.
FAQs on Divisibility Rule by 13
What is the Divisibility Rule of 13?
The divisibility rule of 13 is a set of rules to check if a number can be completely divided by 13, without leaving a remainder. There are 4 ways in which it can be done. They are as follows.
 Rule 1: Group the given number into sets of 3 starting from the right. From the rightmost group of 3 digits apply the subtraction and addition operations alternatively and find the result. If the result is either a 0 or it can be divided by 13 completely without leaving a remainder, then the number is divisible by 13.
 Rule 2: Multiply the ones place digit by 4, and add the product to the rest of the number to the left of the ones place digit. If the resulting number is a 0 or a multiple of 13, then the number is divisible by 13.
 Rule 3: Take the last two digits of a number and subtract it from the product of 4 and the rest of the number. If the resulting number is 0 or a multiple of 13, then we can say that the number is divisible by 13.
 Rule 4: Multiply the number at the ones place by 9 and find the difference between the product obtained and the rest of the number to its left. If the number is 0 or a multiple of 13, then we can say that the given number is divisible by 13.
What is the Divisibility Rule of 13 and 17?
Divisibility Rules of 13 and 17 are as follows:
 For a number to be divisible by 13, we multiply the unit place digit of the number by 4 and add the product to the rest of the number to its left. If the sum is a multiple of 13, then we can say that the given number is divisible by 13.
 For a number to be divisible by 17, we multiply the unit place digit by 5 and subtract the product with the rest of the number to its left. If the resulting difference is 0 or a multiple of 17, then the given number is divisible by 17,
Using the Divisibility Rule of 13, check if 2197 is Divisible by 13?
Let us use the following rule to check the divisibility of 2197 by 13. Multiply the unit place digit of the number by 4 and add the product to the rest of the number to its left. If the sum is a multiple of 13, then we can say that the given number is divisible by 13. The last digit when multiplied by 4 gives 4 × 7 = 28. Add 28 to the rest of the number which is 219, 219 + 28 = 247. We do not know if 247 is a multiple of 13 or not. So we repeat the process again. Multiplying the last digit 7 by 4 we get 4 × 7 = 28. Now, add 28 to the rest of the number which is 24, 24 +28 = 52. 52 is the fourth multiple of 13. Hence, 2197 is divisible by 13.
How do you know if a Big Number is Divisible by 13?
To know if a big number is divisible by 13, let us use one of the rules to find it out. There are four rules to find if a number is divisible by 13. Let us apply the first rule, which states that, "Group the given number into sets of 3 starting from the right, or the ones place. From the rightmost group of 3 digits apply the subtraction and addition operations alternatively and find the result. If the result is either a 0 or it can be divided by 13 completely without leaving a remainder, then the number is divisible by 13." For example, in the number 2,232,516, by applying the subtraction addition and operations alternatively from the rightmost group of 3 digits, we get 516232+2 = 286. 286/13 gives 22 as quotient and 0 as remainder. Therefore, 2,232,516 is divisible by 13.
Using the Divisibility Rule of 13, check if 364 is Divisible by 13?
Let us apply the following rule to check if 364 is divisible by 13. Take the last two digits of a number and find the difference between the number formed by the last two digits and the product of 4 and the rest of the number. If the resulting number is 0 or a multiple of 13, then we can say that the number is divisible by 13. The last two digits are 64. The product of the rest of the digit with 4 is 3 × 4, which is 12. Now, subtracting 12 from 64, we get 52. 52 is a multiple of 13. Hence, 364 is divisible by 13.
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