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Divisibility Rule of 7

Before we dive into the divisibility rule of 7, let’s first understand what divisibility rules are.

Divisibility rules are special tricks or shortcuts in mathematics that help us check whether a given number is divisible by another number without actually performing long division.

For example:

A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
A number is divisible by 5 if it ends in 0 or 5.
A number is divisible by 10 if it ends in 0.

These are simple and instantly recognizable. But not all numbers have such easy rules. Some, like 7, require a few more steps. That’s where the divisibility rule of 7 comes into play.

Table of Contents

What is the Divisibility Rule of 7?
Step by Step Process Divisibility Rule of 7
Divisibility Rule of 7 Examples
Alternate Divisibility Test of 7 Using Multipliers
Why Is the Divisibility Rule of 7 Useful?
Common Misconceptions About the Divisibility Rule of 7
Fun Facts About the Divisibility Rule of 7
Conclusion
Frequently Asked Questions (FAQs)

What is the Divisibility Rule of 7?

A mathematical technique called the divisibility rule of 7 can be used to determine if a number is divisible by 7 without the need for conventional long division.

The divisibility test for 7 is a little more complicated than the rules for smaller numbers because it requires doubling and subtracting, but once understood, it is quite effective.

Step by Step Process Divisibility Rule of 7

Without actually dividing a number by 7, the Divisibility Rule of 7 offers a methodical approach to figuring out if it can be divided equally. Compared to the rules for divisibility by 2, 5, or 10, this rule is a little more complex, but it gets easier with practice.

Here is the step-by-step process to apply the divisibility rule of 7:

Step 1: Take the Last Digit of the Number

Look at the unit's digit (the rightmost digit) of the number.
This digit is what you’ll double in the next step.

Example: For the number 203, the last digit is 3.

Step 2: Double the Last Digit

Multiply the last digit by 2.

Example: 3 × 2 = 6

Step 3: Subtract That Doubled Value from the Remaining Number

Remove the last digit from the number (what's left is called the "remaining number").
Subtract the doubled value (from Step 2) from this remaining number.

Example: Remove 3 from 203 → Remaining number is 20

Subtract: 20 − 6 = 14

Step 4: Check the Result

Check if the result (from Step 3) is divisible by 7.
If it is divisible by 7 (including 0), then the original number is also divisible by 7.

Example: 14 ÷ 7 = 2 → It's a whole number

So, 203 is divisible by 7

Step 5: If the Result is Large, Repeat the Rule

If the result you get after subtracting is still a large number, or you’re unsure, you can apply the same rule again to the new number.

Example: Check 161

Last digit: 1 → Double it = 2
Remaining number: 16
Subtract: 16 − 2 = 14
14 is divisible by 7
So, 161 is divisible by 7

This method is commonly referred to as the divisibility test for 7. Let's break it down with some real examples.

Divisibility Rule of 7 Examples

Example 1: Is 532 divisible by 7?

Let us apply the divisibility rule of 7 step by step:

Step 1: Take the last digit of the number.

In 532, the last digit is 2.

Step 2: Multiply this last digit by 2.

2 × 2 = 4

Step 3: Subtract this result from the remaining number (the part before the last digit).

The remaining number is 53.

So, 53 − 4 = 49

Step 4: Check if the new result (49) is divisible by 7.

Yes, because 7 × 7 = 49.

Conclusion: Since 49 is divisible by 7, we can say that 532 is divisible by 7.

This is a classic divisibility rule of 7 example.

Example 2: Is 905 divisible by 7?

Let’s use the same divisibility test of 7 method.

Step 1: Last digit = 5

Step 2: Double the last digit → 5 × 2 = 10

Step 3: Remaining number = 90

Subtract: 90 − 10 = 80

Step 4: Is 80 divisible by 7?

No. 7 × 11 = 77 and 7 × 12 = 84.

80 lies between 77 and 84 and is not a multiple of 7.

Conclusion: 905 is not divisible by 7.

This example helps understand when the divisibility test for 7 fails.

Example 3: Is 203 divisible by 7?

Let’s test this using the divisibility rule of 7.

Step 1: Last digit = 3

Step 2: Double it → 3 × 2 = 6

Step 3: Remaining number = 20

Subtract: 20 − 6 = 14

Step 4: Check if 14 is divisible by 7.

Yes, 7 × 2 = 14.

Conclusion: Since the final number is divisible by 7, 203 is divisible by 7.

This is a direct and easy-to-follow divisibility rule of 7 example.

Example 4: Is 1015 divisible by 7?

This will involve two rounds of the divisibility test for 7.

Step 1: Last digit = 5

Step 2: Double it → 5 × 2 = 10

Step 3: Remaining part = 101

Subtract: 101 − 10 = 91

Step 4: Check if 91 is divisible by 7.

Yes, because 7 × 13 = 91.

Conclusion: 1015 is divisible by 7.

This shows that even large numbers can be handled easily using the divisibility rule of 7.

Alternate Divisibility Test of 7 Using Multipliers

Another technique for determining divisibility by 7 is particularly helpful for mental math.

This is a different modular arithmetic divisibility test for 7. The digits are multiplied by a repeating pattern, which goes from right to left: 1, 3, 2, 6, 4, 5.

Example: Is 462 divisible by 7?

Digits from right to left: 2, 6, 4

2 × 1 = 2
6 × 3 = 18
4 × 2 = 8

Sum = 2 + 18 + 8 = 28

28 is divisible by 7 (7 × 4 = 28)

So, 462 is divisible by 7.

This is another effective divisibility test of 7.

Why Is the Divisibility Rule of 7 Useful?

It facilitates the simplification of large-number problems.
It helps with algebra and factorization.
In competitive exams, it is commonly utilised.
It improves reasoning and mental computation.
It facilitates error-checking when performing calculations by hand.

The divisibility test of 7 is a valuable skill to learn.

Common Misconceptions About the Divisibility Rule of 7

Misconception: You can check divisibility by 7 just by looking at the last digit.
Correction: The rule requires subtraction and doubling, not just visual checks.
Misconception: The rule works like those for 3 or 9 (adding digits).
Correction: Summing digits is not a part of the divisibility rule of 7.
Misconception: The rule is too complicated to be useful.
Correction: It becomes a quick mental trick after practice.

Fun Facts About the Divisibility Rule of 7

Because of its complexity, the divisibility rule of 7 is not frequently taught in elementary schools, despite the fact that it is highly helpful in advanced math.
These rules, which also appear in number theory and cryptography, are constructed by mathematicians using modular arithmetic.
This rule is one of the few divisibility rules that require actual calculation rather than visual patterns.

Conclusion

The divisibility rule of 7 is a unique and slightly advanced rule that helps determine whether a number is divisible by 7 without using traditional division. It may seem challenging at first, but with practice and examples, it becomes a reliable method.

There are two main techniques:

Subtract twice the last digit from the rest of the number.
Use digit multiplication with modular patterns.

Keep practicing different divisibility rules of 7 examples and try the divisibility test for 7 using both methods to gain confidence.

Frequently Asked Questions (FAQs)

1. What is the divisibility rule of 7?

Double the last digit and subtract it from the remainder of the number to determine if it is divisible by 7. The original number is divisible by 7 if the result is divisible by 7 (or equal to 0).

Example:

For 203:

Double the last digit → 3 × 2 = 6

Subtract from the rest → 20 - 6 = 14

Since 14 is divisible by 7, 203 is divisible by 7.

2. How can I check if a number is divisible by 7?

Use the rule:

Take the last digit, double it, and subtract it from the remaining number.
Repeat the process.
If the result is divisible by 7, then the original number is too.

As an alternative, use long division to divide the number by 7. It is divisible if there is no remainder.

3. Is 1001 divisible by 7?

Yes, 1001 is divisible by 7.

1001 ÷ 7 = 143 (a whole number, no remainder).

4. What is the divisibility trick?

A divisibility trick is a rule or shortcut that allows you to quickly determine whether a number can be divided by another number without having to perform full division. Every number has a unique rule, such as 2, 3, 5, 7, etc.

The trick for 7 is to take the last number and subtract it from the remaining numbers. The original number is divisible by 7 if the result is.