# Divisibility Rule by 7: A Comprehensive Overview

- Author: Noreen Niazi
- Last Updated on: January 14, 2024

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Toggle*Does 2315689 is divisible by 7?*

Can you answer this question within seconds?

No, then you might need to understand the divisibility rule of 7. Divisibility rule by 7 helps us to determine whether number is divisible by 7 are not.

**Divsbility rule by 7, tells us that if we find the difference doubling the number at the unit place and the remaining number, difference is divisible by 7, then given number is divisible by 7.**

So, lets see how to use the divisibility rule to solve a long number, without doing long Divions. This article will focus on the divisibility rule by 7, one of the lesser-known rules. We will discuss what it is, how to use it, and its real-world applications.

## What are Divisibility Rules?

*Can you tell me the number 23456 is divisible by 8.*

How much time you need to find the solution.

Does there’s any shortcut to check whether number is divisible by 8 are not. Yes, there’s. You can determine whether a number is divisible by given number or not by a shortcut method. This shortcut method is known as divisibility rules.

Divisibility rules tell you with in second given number is divisible by others number are not.

For example, the **divisibility rule by 2** tells us that a number is divisible by 2 **if its last digit is even.** Similarly, the** divisibility rule by 5** tells us that a number is divisible by 5** if its last digit is 0 or 5.**

## What is the Divisibility Rule by 7?

Divisibility rule by 7 is the short cut method that tells you that given number is divisible by 7 or not.

A number is **divisible by 7** if and only if the difference between twice the digit in the unit’s place and the remaining digits is divisible by 7.

For example, if we take a number like 532, we need to double the digit in the unit’s place, which is 2, and subtract it from the remaining digits, which gives us 53 – 2(2) = 49. If 49 is divisible by 7, then 532 is also divisible by 7.

## Examples of Numbers Divisible by 7

Now check how divisibility rule by 7 works with some numbers.

**Example 1**

**Check whether 315 is divisible by 7 or not?**

Now use divisibility rule of 7 to identify whether number is divisible by 7 or not.

- Double the digit in the units place, which is 5.
- Now subtract it from the remaining digits, which gives us 31 – 2(5) = 21.
- As 21 is divisible by 7, therefore 315 is also divisible by 7.

**Example 2**

**Check whether 469 is divisible by 7 or not?**

Now use divisibility rule of 7 to identify whether number is divisible by 7 or not.

- Double the digit in the units place, which is 9.
- Now subtract it from the remaining digits, which gives us 46 – 2(9) = 18.
- As 28 is divisible by 7, therefore 469 is also divisible by 7.

## How to Use the Divisibility Rule for 7

Now you are clear about what is divisibility rule of 7 and how does its Works. Move one to next step and check how we can use it.

Here are the three simple step though which you can apply divisibility rule.

**Step :1 **Identify the number at the unit place.

**Step 2: **Double the number and subtract form the reaming digits.

**Step 3: **Check whether the difference is divisible by 7 or not. If its is divisible by 7, then given number is also disvisble by 7.

Let’s take an example to illustrate this.

**Example**

**Check whether 728 is divisible by 7. **

**Step 1:** Digit at unit place is 8.

**Step 2:** By doubling 8 we get 16.

**Step 3:** Subtracting 16 from the remaining digits, we get 72 – 16 = 56.

Since 56 is divisible by 7, we know that 728 is also divisible by 7.

## Understanding the Math Behind the Rule

The divisibility rule by 7 may seem like magic, but it is based on sound mathematical principles. To understand why it works, we need to delve deeper into the properties of numbers.

**Every number can be expressed as a sum of its digits multiplied by powers of 10.**

**For example**,

We can write 532 can

5 x 100 + 3 x 10 + 2 x 1.

Now we can arrange the expression as

5 x (99 + 1) + 3 x (9 + 1) + 2 x 1,

which simplifies to

5 x 99 + 3 x 9 + 7.

Notice that the sum of the digits of the original number, **5 + 3 + 2 = 10**, is the same as the remainder we get when we divide the expression by 9 (7 in this case).

Now, let’s apply this concept to the divisibility rule by 7.

If we take a number like 532, we can express it as

5 x 100 + 3 x 10 + 2 x 1.

Doubling the digit in the unit’s place and subtracting it from the remaining digits, we get

(5 x 10 – 2) x 10 + 3 x 1.

Rearranging this expression, we get (5 x 9 + 3) x 10 + 7.

Notice that the sum of the digits of the original number, 5 + 3 + 2 = 10, is the same as the remainder we get when we divide the expression by 7 . This is why the divisibility rule by 7 works.

## Divisibility Rules for Other Numbers

The divisibility rule by 7 is just one of many divisibility rules. Some of the other popular rules are:

- Divisibility rule by 2: A number is divisible by 2 if its last digit is even.
- Divisibility rule by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility rule by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- Divisibility rule by 5: A number is divisible by 5 if its last digit is 0 or 5.
- Divisibility rule by 6: A number is divisible by 6 if it is divisible by 2 and 3.
- Divisibility rule by 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
- Divisibility rule by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
- Divisibility rule by 10: A number is divisible by 10 if its last digit is 0.

## Practice Problems to Test Your Knowledge

Now that we have discussed the divisibility rule by 7 in detail, let’s test our knowledge with some practice problems. Try to solve these problems using the divisibility rule by 7.

- Is the number 357 divisible by 7?
- Is the number 924 divisible by 7?
- Is the number 8756 divisible by 7?
- Is the number 1234 divisible by 7?
- Is the number 7777 divisible by 7?

You can check your answers at the end of this article.

## Tricks to Remember the Rule

Remembering the divisibility rule by 7 can be tricky, especially if you use it sparingly. Here are a few tricks that can help you remember the rule:

- Notice that twice the digit in the unit’s place is the same as subtracting 5 times the digit in the unit’s place from the number formed by the remaining digits.
**For example**, if we take the number 532, we can express it as 53 x 10 + 2 x 1. Notice that 2 x 2 = 4 is the same as subtracting 5 x 2 = 10 from 53.

- If you are testing a large number for divisibility by 7, you can break it down into smaller parts.
**For example**, if we take the number 1234567, we can break it down into 123 x 10000 + 4567. Applying the divisibility rule by 7 to this expression, we get (123 x 2 – 456) x 1000 + 7. Notice that the expression in the parentheses is divisible by 7, so we only need to test whether 7 divides into 7, which it does.

## Real-World Applications of the Divisibility Rule by 7

You may wonder why you would ever need to use the divisibility rule by 7 in real life. While it may not be a commonly used rule, it does have some practical applications. For example, it can be used to check whether a credit card number is valid. Credit card numbers are typically 16 digits long and follow a specific pattern. The last digit is the check digit, calculated using a formula involving the other digits in the number. One of the steps in this formula is to check whether a certain portion of the number is divisible by 7.

## Conclusion

In conclusion, the divisibility rule by 7 is a useful shortcut that can save us a lot of time when performing calculations. By understanding the concept behind the rule and practicing with some examples, you can become proficient in using it. Remember to use tricks like breaking down large numbers into smaller parts and noticing patterns to make the process easier. While the divisibility rule by 7 may be used infrequently, it does have practical applications in fields like banking and finance. Keep practicing and honing your math skills; you’ll be amazed at what you can accomplish.

## Practice Problems Answers

- Yes, 357 is divisible by 7.
- No, 924 is not divisible by 7.
- Yes, 8756 is divisible by 7.
- No, 1234 is not divisible by 7.
- Yes, 7777 is divisible by 7.