Divisibility Rule of 13 with Examples and FAQ – Mindspark

A number is divisible by 13 if it leaves zero as the remainder when divided by 13. Here we will understand the divisibility rule of 13 with some examples.

The Divisibility Rule of 13

The divisibility rule of 13 helps us know if a number can be divided by 13 without doing the long division process. There are 4 divisibility rules of 13. Let’s know about them one by one:

First Rule:

Starting from the right (ones place), make a group of the given numbers into sets of 3.

From the rightmost group, subtract the first set from the second set and then add the remaining digit(s). 

Now calculate the results. If the result is either 0 or a number divided by 13 wholly, the given number will be divisible by 13.

For example: In the number 1153854 from the rightmost side, we have:

First set of 3 numbers = 854

Second set = 153, and the remaining digit = 1.

Subtracting the first and second sets, we get 854 – 153 = 701

Now add the remaining digit, we get 701 + 1 = 702

The result 702 is divisible by 13.

Therefore, 1153854 is divisible by 13.

Second Rule:

Multiply the last digit of the given number by 4 and add the product with the rest of the number to the left of the last digit. 

If the resulting number is either a 0 or divisible by 13, the given number will be divisible by 13.

For example: In the number 533, the unit place digit is 3. 

Multiplying it by 4, we get (3 x 4), which is 12. 

Adding 12 to the rest of the digits to the left, we get 53 + 12 = 65. 

Since 65 is a multiple of 13.

Hence 533 is divisible by 13.

Third Rule:

Take the number formed by the last two digits of the given number. 

Multiply the rest of the number to the left of these two digits by 4. 

Now subtract the number formed by the last two digits from the product. 

If the resulting number is either 0 or a multiple of 13, we can confirm that the number is divisible by 13. 

For example: In the number 611, the last two digits are 11. 

Product of 4 and the rest of the number (6) is 6 x 4 = 24. 

On subtracting them, we get 24 – 11, which is 13 (a multiple of 13). 

Therefore, 611 is divisible by 13.

Fourth Rule:

Multiply the last digit by 9. Now subtract the product and the rest of the number to the left of the last digit. If the difference is either 0 or a multiple of 13, we can confirm that the given number is divisible by 13.

For example: In the number 676, multiplying the last digit (6) with 9, we get 6 x 9, which is 54. On subtracting this from the rest of the number, which is 67, we get 67 – 54, which is 13. 

Since 13 is a multiple of 13, we can say that 676 is divisible by 13.

Note – If the given number is large. If we are unsure if the resultant number obtained from any of the above rules is a multiple of 13, we repeat the divisibility test until we reach a number that can be checked easily from the multiplication table of 13 whether it’s divisible by 13 or not.

For Example: Using the divisibility rule of 13, find if 2782 is divisible by 13 or not.

Follow the below steps to check for the divisibility of 13 – 

• Multiply the ones place digit by 9. We get (9 × 2) = 18.
• Subtract 18 from 278. 278 – 18 = 260. Is 260 a multiple of 13? We are not sure. Let us repeat the process.
• Multiply the last digit of 260 by 9. We get 0 × 9 = 0.
• Find the difference between 0 and the rest of the number (26), 26 – 0 = 26.

26 is divisible by 13. Therefore, 2782 is divisible by 13.

Example:

1. Using the divisibility rule of 13, find if 59371 is divisible by 13 or not?

Multiply the last digit of 59371 by 9, which is 9 × 1 = 9.

The remaining number left = 5937

Subtract 9 from 5937, 5937 – 9 = 5928. 

We are not sure if 5928 is a multiple of 13. Let us repeat the process.

Multiply the last digit of 5928 by 9, which is 8 × 9 = 72.

Find the difference between 592 and 72, which is 592 – 72 = 520. 

We are still not sure if 520 is a multiple of 13. Let us repeat the process.

Multiply the last digit of 520 by 9, which is 0 × 9 = 0.

Find the difference between 52 and 0, which is 52, a multiple of 13. 

Therefore, 59371 is divisible by 13.

2. Check if 3063580 is divisible by 13 or not.

After grouping the number into sets of 3, 

First set from the rightmost = 580

Second set = 063

Remaining digit = 3

Subtracting the first and second sets, we get 580 – 63 = 517

Adding the remaining digit, we get 517 + 3 = 520

When 520 is divided by 13, we get the remainder as 0. Hence, 3063580 is divisible by 13.