Divisibility Rule of 7 with Examples (original) (raw)
Last Updated : 23 Jul, 2025
**Divisibility rules are simple mathematical shortcuts that help determine whether a number can be divided by another number without performing the actual division. In the case of 7, the divisibility rule for 7 helps us quickly determine if a number is divisible by 7 without needing to perform long division.
Divisibility Rule of 7:
- **Take the last digit (unit digit) of the number and double it.
- **Subtract this doubled value from the rest of the digits in the number.
- **Check the result: If the difference is divisible by 7 (or is 0), then the original number is divisible by 7.
More Examples on Divisibility by 7 Rule
Here are a few examples of numbers divisible by 7, applying the divisibility rule:
**For 196:
- Take the last digit (6), and double it to get 12.
- Subtract 12 from 19 (remaining digits): 19 − 12 = 7.
- Since 7 is divisible by 7, 196 is divisible by 7.
**For 357:
- Double the last digit (7) to get 14.
- Subtract 14 from 35: 35 − 14 = 21.
- Since 21 is divisible by 7, 357 is divisible by 7.
Divisibility Rule of 7 Proof
A general number N can be written as:
N = 10^n a_n + 10^{n-1} a_{n-1} + \cdots + 10 a_1 + a_0
Here, a_n, a_{n-1}, \dots, a_1, a_0 are the digits of the number. We want to show that N is divisible by 7, i.e., N = 7k for some integer k.
We can factor out 10 from all terms except the last one, giving:
N = 10 \left( 10^{n-1} a_n + 10^{n-2} a_{n-1} + \cdots + 10 a_2 + a_1 \right) + a_0
Now, to introduce the rule of subtracting twice the last digit, we **add and subtract 20 a_0
N = 10 \left( 10^{n-1} a_n + 10^{n-2} a_{n-1} + \cdots + 10 a_2 + a_1 \right) + 20 a_0 - 20 a_0 + a_0
This simplifies to:
N = 10 \left( 10^{n-1} a_n + 10^{n-2} a_{n-1} + \cdots + 10 a_2 + a_1 - 2 a_0 \right) + 21 a_0
Now, notice that:
- 21 \equiv 0 \mod 7 so the term 21 a_0 contributes nothing to the remainder modulo 7.
- We only need to check whether 10 \left( 10^{n-1} a_n + 10^{n-2} a_{n-1} + \cdots + a_1 - 2 a_0 \right) \equiv 0 \mod 7
10 (\overline{a_na_{n-1}......a_2a_1} - 2a_0) \equiv 0 (mod 7)
since 10 ≡ 3 (mod 7), for N to be divisible by 7, \overline{a_na_{n-1}......a_2a_1} - 2a_0) \equiv 0 (mod 7)
Divisibility Rule of 7 Solved Questions
**Example 1: Check if the given number is divisible by 7 or not: 458409
**Solution:
Let us check if the given number, 458409 is divisible by 7 or not using the following steps:
- **Step 1: We first take the last digit and multiply it by 2. So,(9 × 2 = 18).
Subtract 18 with the rest of the number, which is 45840. So, 45840 - 18 = 45822.
We are not sure if 45822 is a multiple of 7.- **Step 2: We repeat the same process again with 45822.
Multiply the last digit by 2. So, (2 × 2 = 4).
Subtract 4 with the rest of the number, which is 4582. So, 4582 - 4 = 4578.
We are not sure if 4578 is a multiple of 7.- **Step 3: Let us repeat the process again with 4578.
Multiply the last digit by 2. So, (8 × 2 = 16).
Subtract 16 with the rest of the number, which is 457. So, 457 - 16 = 441.
We are not sure if 441 is a multiple of 7.- **Step 4: Let us repeat the process again with 441.
Multiply the last digit by 2. So, (1 × 2 = 2).
Subtract 2 with the rest of the number, which is 44. So, 44 - 2 = 42.
42 is the sixth multiple of 7.Therefore, we can confirm that 458409 is divisible by 7.
**Example 2: Is 154 divisible by 7?
**Solution:
The last digit in the given number 15 4 (unit digit ) is 4.
We now use the given number without the last digit which is 15.
Subtract twice the last digit 4 from 15:
15 - 2 (4) = 15 - 8 = 7The result 7 is a multiple of 7 and therefore 154 is divisible by 7.
**Example 3: Consider the number: 308. Check if it is divisible by 7.
**Solution:
Following the rule:
Double of the last digit =16
Subtracting the result from the rest of the number; 30-16 =1414 is a multiple of 7, hence the number is divisible by 7.
**Example 4: Which of the following numbers is divisible by 7?
- **171
- **119
- **107
- **383
**Solution:
The correct answer is **option (b) 119.
**Explanation:
(a) 171
Step 1: Double the unit digit = 1 x 2 = 2
Step 2: Difference = 17 – 2 = 1515 is not a multiple of 7, and hence 171 is not divisible by 7.
****(b) 119**
Step 1: Double the unit digit = 9 x 2 = 18
Step 2: Difference = 11 – 18 = -7, which is a multiple of 7Hence, 119 is divisible by 7.
(c) 107
Step 1: Double the unit digit = 7 x 2 = 14
Step 2: Difference = 10 – 14 = -4, which is not a multiple of 7.Hence, 107 is not divisible by 7.
(d) 383
Step 1: Double the unit digit = 3 x 2 = 6
Step 2: Difference = 38 – 6 = 32, which is not a multiple of 7.Thus, 383 is not divisible by 7.
**Example 5: Check whether a number 449 is divisible by 7.
**Solution:
Given number = 449.
To check whether a number 449 is divisible by 7, follow the below steps.
Step 1: Double the unit digit = 9 x 2 = 18
Step 2: Take the difference between the remaining part of the given number and the result obtained from step 1. (i.e., 18)
= 44 – 18
= 26, which is not a multiple of 7.Hence, the given number 449 is not divisible by 7.
Divisibility Rule of 7 Worksheet
**1. Determine whether the number 203 is divisible by 7.
**2. Is the number 1,218 divisible by 7? Show your calculation.
**3. Find out if 5,643 is divisible by 7 using the divisibility rule.
**4. Check if 2,118 is divisible by 7 and explain your reasoning.
**5. Use the divisibility rule to determine if 9,374 is divisible by 7.
Conclusion
In this article, we discovered the method for determining if a number is divisible by 7 and the rule for divisibility by 7 is an important guideline that applies to all numbers. When divided by 7 there should be no remainder left after the division. This holds true because whenever we divide a number by 7 the outcome is consistently an even number. Keep practicing this divisibility rule to get a better grasp at the concept.
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