Are you struggling with checking if a number is divisible by 13? Don’t worry; there is a simple trick to help you. In this article, we will discuss the Divisibility Rule for 13, a quick and easy way to determine if a number is divisible by 13 without using a calculator. Let’s get started!

## What is the Divisibility Rule for 13?

The Divisibility Rule for 13 states that if you subtract four times the digit in the unit’s place from the number formed by the remaining digits, the result should be divisible by 13. If the result is zero, the number is divisible by 13.

For example, let’s take the number 247. The digit in the unit’s place is 7. Now, subtracting four times 7 from the number formed by the remaining digits (24) gives us 24 – 28 = -4. Since -4 is not divisible by 13, 247 is not divisible by 13.

## How to Apply the Divisibility Rule for 13?

Let’s take another example to understand the application of the Divisibility Rule for 13. Consider the number 1,014.

- Remove the digit in the unit’s place from the number. This gives us 101.
- Multiply the digit in the unit’s place (4) by 4. This gives us 16.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 101 – 16 = 85.
- Repeat steps 1 to 3 until the result obtained is either zero or a number smaller than 13.

**Let’s apply these steps again.**

- Remove the digit in the unit’s place from the number. This gives us 8.
- Multiply the digit in the unit’s place (5) by 4. This gives us 20.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 8 – 20 = -12.
- Since -12 is not divisible by 13, we need to repeat the steps.
- Remove the digit in the unit’s place from the number. This gives us 1.
- Multiply the digit in the unit’s place (8) by 4. This gives us 32.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 1 – 32 = -31.
- Since -31 is not divisible by 13, we need to repeat the steps.
- Remove the digit in the unit’s place from the number. This gives us 3.
- Multiply the digit in the unit’s place (1) by 4. This gives us 4.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 3 – 4 = -1.
- Since -1 is not divisible by 13, we need to repeat the steps.
- Remove the digit in the unit’s place from the number. This gives us 0.
- Multiply the digit in the unit’s place (3) by 4. This gives us 12.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 0 – 12 = -12.
- Since -12 is not divisible by 13, we need to repeat the steps.

We can see that we are repeating the same steps again and again, and the result obtained is not changing. Therefore,

We can see that we are repeating the same steps again and again, and the result obtained is not changing. Therefore, we can conclude that the number 1,014 is not divisible by 13.

## Why is the Divisibility Rule for 13 Useful?

The Divisibility Rule for 13 is a handy tool for quickly checking if a number is divisible by 13. It is particularly useful when you do not have a calculator or when the number is too large to use trial division.

The rule is also helpful when solving math problems that involve division by 13, such as finding the remainder of a division or determining if a fraction is simplified.

## Tips for Using the Divisibility Rule for 13

Here are some tips to make using the Divisibility Rule for 13 more manageable:

- The rule applies only to whole numbers.
- Always start from the right and work your way left when using the rule.
- When subtracting, you can add or subtract multiples of 13 without changing the result.
- If the number formed by the remaining digits is small, it may be quicker to use trial division instead of the rule.

## Examples of Using the Divisibility Rule for 13

Let’s look at some examples to see how the Divisibility Rule for 13 works in practice:

**Example 1: Is 2,345 divisible by 13?**

- Remove the digit in the unit’s place from the number. This gives us 234.
- Multiply the digit in the unit’s place (5) by 4. This gives us 20.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 234 – 20 = 214.
- Repeat steps 1 to 3 until the result obtained is either zero or a number smaller than 13.
- Remove the digit in the unit’s place from the number. This gives us 21.
- Multiply the digit in the unit’s place (4) by 4. This gives us 16.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 21 – 16 = 5.
- Since 5 is not divisible by 13, the number 2,345 is not divisible by 13.

**Example 2: Is 6,826 divisible by 13?**

- Remove the digit in the unit’s place from the number. This gives us 682.
- Multiply the digit in the unit’s place (6) by 4. This gives us 24.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 682 – 24 = 658.
- Repeat steps 1 to 3 until the result obtained is either zero or a number smaller than 13.
- Remove the digit in the unit’s place from the number. This gives us 65.
- Multiply the digit in the unit’s place (8) by 4. This gives us 32.
- Subtract the result obtained in step 2 from the number obtained in step 1. This gives us 65 – 32 = 33.
- Since 33 is divisible by 13, the number 6,826 is divisible by 13.

## Conclusion

In conclusion, the Divisibility Rule for 13 is an efficient method for determining if a number is divisible by 13. It saves time and effort compared to traditional methods like trial division. By following the simple steps outlined in this article, you can quickly determine whether a number is divisible by 13 or not.

## FAQs

### Does the Divisibility Rule for 13 work for all whole numbers?

No, the rule only applies to whole numbers.

### What happens if the result obtained from the rule is negative?

If the result obtained from the rule is negative, you need to repeat the steps until you get a non-negative number.

### Can I use the Divisibility Rule for 13 to find the quotient and remainder of a division?

Yes, you can use the rule to find the remainder of a division. To find the quotient, you need to divide the number obtained in the last step by 13.

### Can the Divisibility Rule for 13 be generalized to other numbers?

Yes, there are divisibility rules for other numbers, such as 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12.

### Is the Divisibility Rule for 13 commonly used in real-world applications?

While the rule may not be used in everyday life, it is a helpful tool for solving mathematical problems and checking divisibility quickly.

Now that you know how to use the Divisibility Rule for 13, you can save time and effort when solving problems that involve division by 13. Happy calculating!