Divisibility Rule Of 13 - Methods, Examples - Cuemath

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 when 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 divisible by 13.

1.	What is the Divisibility Rule of 13?
2.	Divisibility Rule of 13 for Large Numbers
3.	Divisibility Rule of 13 and 14
4.	Divisibility Test of 13 and 17
5.	FAQs on Divisibility Rule of 13

What is the Divisibility Rule of 13?

The divisibility rule of 13 helps us to determine if a number is divided by 13 without leaving any remainder. 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.

Divisibility Rule of 13 with Examples

There are 4 divisibility rules to check if a number is divisible by 13 or not. Let us study these 4 rules in detail with examples.

Divisibility by 13 Rule 1:

Group the given number into sets of 3 starting from the right, or the units 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 502 - 139 + 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 units place digit. If the resulting number is a 0 or a multiple 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 × 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 × 4, which is 20. On subtracting them, we get 20-20, which is 0. Therefore, 520 is divisible by 13. This method of divisibility by 13 can be easily applied and very effective with three-digit 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 × 9, which is 27. On subtracting this from the rest of the number, which is 79, we get 79 - 27, 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 5-digit 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). 15 -12 = 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 units place digit of a number by 5 and find the difference between the product and 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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