Divisibility rule has come into the light to solve large numbers of division problems easily. This rule is used to check whether an integer (dividend) can be completely divided by any other integer (divisor) or not. Suppose if the number is very large and has to be divided by 13, and if we apply the division method to check its divisibility, it will take more time. Therefore, the divisibility rules for 13 were introduced.

Apart from 13, we have divisibility rules for 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on. For example, in case of divisibility by 2 rule, 82 is a number which is completely divided by 2 as the digit in unit place is even and all the even numbers are divided by 2. But if the number in unit place is odd, then it cannot be completely divided by 2. By this simple rule, we can configure if any number is completely divided by 2 or not.

Let us learn more about divisibility rules for number 13, by definitions and examples.

What are the Divisibility Rules For 13?

There are basically 4 types of divisibility rules for 13. Go through the divisibility rules for 13 with examples given below.

Divisibility of a number by 13 Rule 1

Rule: For a given number, form alternating sums of blocks of three numbers from the right and moving towards the left. Suppose ( \(n_{1}n_{2}n_{3}n_{4}n_{5}n_{6}….\) ) is a number N, then if the number formed by the alternative sum and difference of blocks of 3-3 digits from right to the left is divisible by 13 then N is divisible by 13. Here, one should apply the subtraction first and then addition. See the example below to understand.

Example: 

Check whether the 2,453,674 is divisible by 13.

Solution:

By applying Rule 1,

674 – 453 + 2 = 223 is not divisible by 13

Therefore, 2,453,674 also is not divisible by 13.

Divisibility of a number by 13 Rule 2

Rule: If a number N is given, then multiply the last digit of N with 4 and add it to the rest truncate of the number. If the outcome is divisible by 13 then the number N is also divisible by 13.

Repeat the process until it is necessary to decide whether the number is divisible by 13. This method is suitable for checking the divisibility of large numbers by 13.

Example: 

Let a number be 650. Find whether it is divisible by 13.

Solution:

By applying Rule 2,

650: 65 + (0 × 4) = 65 and number 65 is divisible by 13 and gives the divisor as 5.

Therefore, 650 is also divisible by 13.

Divisibility of a number by 13 Rule 3

Rule: For a number N, to check whether it is divisible by 13 or not, subtract the last 2 digits of the number N from the 4 times multiple of the rest of the number. However, it is the most accurate method when the given number is a three-digit number.

Example: 

Consider the number 728. Check whether it is divisible by 13 or not.

Solution:

By applying divisibility by 13 rule, we get,

728: (7 × 4) – 28 = 28 – 28 = 0 and number 0 is divisible by 13 giving the result as 0.

Divisibility of a number by 13 Rule 4

Rule: Multiply the last digit, i.e. unit digit by 9 of a number N and subtract it from the rest of the number. If the outcome is divisible by 13 then the number N is divisible by 13.

Example: 

Check whether the number 858 is divisible by 13.

Solution:

By applying rule 4,

858: 85 – (8 × 9) = 13, and 13 is divisible by 13.

Therefore, 858 is divisible by 13.

Practice Problems

1. Check whether the number 451728 is divisible by 13.
2. Prove that the number 61828 is divisible by 13.
3. Check the divisibility for the given numbers by 13:
   (i) 715416               (ii) 2547               (iii)  4602

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