## Divisibility Rule of 7

The divisibility rule for 7, while less straightforward than rules for smaller numbers, is a useful mathematical tool for identifying divisibility within the integer subset of rational number. This rule involves taking the last digit of the number, doubling it, and then subtracting this product from the remaining leading truncated number. Repeating this process as needed can indicate divisibility by 7 when the resulting smaller number is known to be divisible by 7. In algebra and arithmetic, mastering this rule enhances one’s ability to simplify and factorize expressions and solve equations efficiently. The rule applies specifically to integers, aiding significantly in tasks involving addition, subtraction, multiplication, and division.

Download Proof of Divisibility Rule of 7 in PDF

## What is Divisibility Rule of 7?

## Proof of Divisibility Rule of 7

Download Proof of Divisibility Rule of 7 in PDF

To prove the divisibility rule for 7 using a specific number, let’s use the number 352. Here’s the step-by-step explanation of how the rule applies to this number:

### Step 1: Consider the number 352.

### Step 2: Apply the rule for divisibility by 7:

To use the rule, take the last digit of the number, double it, and subtract this product from the rest of the number.

### Step 3: Separate the last digit from the rest:

352→35 and 2

### Step 4: Double the last digit:

2×2 = 4

### Step 5: Subtract the doubled digit from the remaining number:

35−4 = 31

### Step 6: Check the result:

Now, determine if 31 is divisible by 7. Since 31 is not divisible by 7, we repeat the process if necessary or directly assess the smaller number.

### Step 7: Apply the rule again if unclear (not needed in this example):

For simplicity, we can state that since 31 is not divisible by 7, it is clear from this first step. Alternatively, further application could be made by repeating steps with smaller results from further operations, but it’s evident here.

### Step 8: Conclusion:

Since the result, 31, is not divisible by 7, the original number 352 is also not divisible by 7.

## Divisibility Rule of 7 and 11

### Divisibility Rule for 7

- Take the last digit of the number, double it.
- Subtract the doubled digit from the rest of the number.
- If the result is 0 or divisible by 7, then the original number is divisible by 7.
- If the result is still large or unclear, repeat the process with the new number.

### Example:

For the number 203:

- Last digit is 3, doubled is 6.
- Subtract 6 from 20, which gives 14.
- Since 14 is divisible by 7, 203 is also divisible by 7.

### Divisibility Rule for 11

- Add and subtract the digits of the number in an alternating pattern.
- Start from the rightmost digit: add, then subtract the next, and so on.
- If the result is 0 or divisible by 11, then the original number is divisible by 11.

### Example:

For the number 2728:

- Calculation: 8 – 2 + 7 – 2 = 11
- Since 11 is divisible by 11, 2728 is also divisible by 11.

## Divisibility Rule of 7 and 13

### Divisibility Rule for 7:

- Take the last digit of the number, double it.
- Subtract the doubled digit from the rest of the number.
- If the resulting number is 0 or divisible by 7, then the original number is divisible by 7.
- If the result is still large, repeat the process with the new number until you can easily determine its divisibility.

### Example:

Consider the number 161:

- Last digit is 1, doubled is 2.
- Subtract 2 from 16, which gives 14.
- Since 14 is divisible by 7, 161 is also divisible by 7.

### Divisibility Rule for 13:

- Take the last digit of the number, multiply it by 4.
- Add this product to the rest of the number.
- If the resulting number is 0 or divisible by 13, then the original number is divisible by 13.
- If the result is still large, repeat the process with the new number until you can easily determine its divisibility.

### Example:

Consider the number 455:

- Last digit is 5, multiplied by 4 is 20.
- Add 20 to 45, which gives 65.
- Since 65 is divisible by 13 (65 ÷ 13 = 5), 455 is also divisible by 13.

Divisibility Rule of 7 Examples

### Number 203

**Step 1:**Take the last digit (3) and double it (3 x 2 = 6).**Step 2:**Subtract the doubled digit (6) from the rest of the number (20 – 6 = 14).**Step 3:**Since 14 is divisible by 7, the original number (203) is also divisible by 7.

### Number 352

**Step 1:**Take the last digit (2) and double it (2 x 2 = 4).**Step 2:**Subtract the doubled digit (4) from the rest of the number (35 – 4 = 31).**Step 3:**Since 31 is not divisible by 7, the number 352 is not divisible by 7.

### Number 1617

**Step 1:**Take the last digit (7) and double it (7 x 2 = 14).**Step 2:**Subtract the doubled digit (14) from the rest of the number (161 – 14 = 147).**Step 3:**Check if the result (147) is divisible by 7. It is, since 147 ÷ 7 = 21.**Step 4:**Since 147 is divisible by 7, the original number (1617) is also divisible by 7.

### Number 994

**Step 1:**Take the last digit (4) and double it (4 x 2 = 8).**Step 2:**Subtract the doubled digit (8) from the rest of the number (99 – 8 = 91).**Step 3:**Check if the result (91) is divisible by 7. It is, since 91 ÷ 7 = 13.**Step 4:**Since 91 is divisible by 7, the original number (994) is also divisible by 7.

## FAQs

## Can you apply the divisibility rule for 7 to any integer?

Yes, the rule applies to any integer, whether positive or negative. It helps in quickly determining if the number is divisible by 7 without performing complete division.

## Is the divisibility rule for 7 applicable repeatedly?

Yes, if the result of the initial application is still too large to assess easily, you can repeatedly apply the rule to the new result until a clear determination can be made.

## Why do we double the last digit in the divisibility rule for 7?

Doubling the last digit and subtracting it from the rest of the number simplifies the number to a smaller one that can more easily be checked for divisibility by 7. This particular operation alters the number in a way that maintains its divisibility characteristics regarding 7.

## What happens if the result of applying the rule is a negative number?

If you get a negative result, you can continue applying the rule or check if the absolute value of the result is divisible by 7. The sign does not affect divisibility by 7.

## Can the divisibility rule for 7 help in simplifying fractions?

Absolutely, knowing whether the numerator or denominator is divisible by 7 can help in reducing fractions to their simplest form.

## How does the divisibility rule for 7 differ from those of other numbers like 3 or 5?

The rule for 7 involves more steps and requires subtractive manipulation, unlike simpler rules for 3 or 5, which are based on the sum or the last digit of the number, respectively.

## What is an example of applying the divisibility rule for 7 to a large number?

For a large number like 1729, you would take 9 (double it to 18), subtract from 172 to get 154. Since 154 is clearly divisible by 7 (154 ÷ 7 = 22), 1729 is also divisible by 7.

## How can this rule be taught to students learning divisibility?

Introduce the rule step-by-step with examples and encourage practice with both small and large numbers. Visual aids or manipulatives can help demonstrate the subtractive process clearly.

## Are there any limitations to the divisibility rule for 7?

The main limitation is that it might require multiple iterations to reach a manageable number, especially if the initial number is very large, making it potentially time-consuming compared to divisibility checks for smaller numbers like 2 or 5.