Divisibility Rule of 7
The divisibility rule of 7 states that for a number to be divisible by 7, the last digit of the given number should be multiplied by 2 and then subtracted with the rest of the number leaving the last digit. If the difference is 0 or a multiple of 7, then it is divisible by 7. The 'Divisibility rule' or ' Divisibility test of 7' helps us to check if a number is completely divisible by 7 without actually doing the division.
What is the Divisibility Rule of 7?
The divisibility rule of 7 helps us to check if a number can be completely divided by 7 without any remainder. Divisibility means checking if a number is divisible by another number without actually dividing the number. Usually, we perform the division arithmetic operation to know this. However, the divisibility rule of 7 has a shortcut method to find if a number is divisible by 7. The divisibility rule of 7 picks the last digit of a number, multiplies it by 2, and subtracts it with the rest of the number to its left. We check to see if the difference is a 0 or a multiple of 7 to confirm that it is completely divisible by 7.
Let us now learn how to check if a number is divisible by 7. As we have already discussed, a number is perfectly divisible by another number if it does not leave any remainder and a quotient is a whole number. The same rule applies to the divisibility by 7. Observe the following figure to learn the divisibility rules for 7.
Divisibility Rule of 7 For Large Numbers
It is easy to check the divisibility rule of 7 for smaller numbers. However, for larger numbers, we perform the divisibility test of 7. In the case of larger numbers, we repeat the process of applying the divisibility test again and again until we are sure that the number is divisible by 7.
Let us take a 6digit number, 458409.
 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.
 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.
 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.
 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.
Observe the following figure to check if 2455 is divisible by 7.
From the figure, we conclude that 2455 is not divisible by 7. The same rules can be applied for numbers with more than 4digits also.
Divisibility Rule of 7 and 13
Divisibility rules help us to check if a number is completely divisible by another number without actually doing the division. The divisibility rules of 7 and 13 are different. As per the divisibility rule of 7, the last digit is multiplied by 2, and the product is subtracted from the rest of the number. If the difference is 0 or a multiple of 7, then we say that the given number is divisible by 7.
There are four methods by which we check the divisibility of a number by 13. Here, let us discuss one of these methods. As per one of the divisibility rules of 13, we multiply the last digit by 4, and add the product to the rest of the number. If the sum is a multiple of 13, then the number is divisible by 13. If the number is large, we repeat the same process again. Let us understand this with an example.
Let us check if the number 442 is divisible by 7 and 13.
Divisibility of 442 by 7  Divisibility of 442 by 13 

Multiply the last digit by 2. 2 × 2 = 4 
Multiply the last digit by 4. 2 × 4 = 8 
Subtract the product (4) from the rest of the number(44). 44  4= 40 
Add the product (8) to the rest of the number (44) 44 + 8 = 52 
Is 40 a multiple of 7? No, hence, 442 is NOT divisible by 7.  Is 52 a multiple of 13? Yes, hence, 442 is divisible by 13. 
Here, we observe that 442 is NOT divisible by 7 but divisible by 13.
Divisibility Rule of 7 and 8
The divisibility rules of 7 and 8 are different. The divisibility rule of 7 states that the digit at the units place should be multiplied by 2, then the product needs to be subtracted from the rest of the number. If this difference results in a 0 or a multiple of 7, then the number is said to be divisible by 7. For a number to be divisible by 8, we check if the last three digits can be divided by 8 without leaving a remainder or the last three digits are 0.
Let us work out the divisibility rule of 7 and 8 for the number 742.
Divisibility of 742 by 7  Divisibility of 742 by 8 

Multiply the last digit by 2. (2 × 2 = 4)  Check if the last three digits are 0 or a number divisible by 8. 
Subtract the product (4) from the rest of the number(74) 74  4 = 70 
The last three digits are 742. Here, 742/8 leaves a quotient of 92 and a remainder of 6. 
Is 70 a multiple of 7? Yes, hence, 742 is divisible by 7.  Therefore, 742 is NOT divisible by 8. 
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Divisibility Rule of 7 Examples

Example 1: Using the divisibility rule of 7, find out if 2415 is divisible by 7?
Solution:
Let us apply the divisibility rule of 7 to 2415 to check whether it is divisible by 7 or not.
Step 1: Multiply the last digit (5) by 2. The product is 10.
Step 2: Subtract the product (10) from the rest of the number, which is 241. (241  10 = 231)
Step 3: We do not know if 231 is a multiple of 7. So, let us go back to step 1 with the number 231.
Step 4: Multiply the last digit (1) by 2. The product is 2.
Step 5: Subtract it from the rest of the number, which is 23. (23  2 = 21)
Step 6: Is 21 divisible by 7? Yes, so we can conclude that 2415 is divisible by 7. 
Example 2: Robin wants to know if 3216 is divisible by 7. Can you help him?
Solution:
Let us apply the divisibility rule of 7 to 3216 to check whether it is divisible by 7 or not.
Step 1: Multiply the last digit (6) by 2. The product is 12.
Step 2: Subtract the product (12) from the rest of the number, which is 321. (321  12 = 309)
Step 3: We do not know if 309 is a multiple of 7. So, let us go back to step 1 with the number 309.
Step 4: Multiply the last digit (9) by 2. The product is 18.
Step 5: Subtract it from the rest of the number, which is 30. (30  18 = 12)
Step 6: Is 12 divisible by 7? No, so we can conclude that 3216 is NOT divisible by 7. 
Example 3: Using the divisibility test of 7, check if 195 is divisible by 7.
Solution:
Let us apply the divisibility test of 7 to 195 to check whether it is divisible by 7 or not.
Step 1: Multiply the last digit (5) by 2. The product is 10.
Step 2: Subtract the product (10) from the rest of the number, which is 19. (19  10 = 9)
Step 3: 9 is NOT a multiple of 7.
Step 4: Therefore, 195 is NOT divisible by 7.
FAQs on Divisibility Rule of 7
What is the Divisibility Rule of 7?
As per the divisibility rule of 7, the last digit of the given number is multiplied by 2, and the product is subtracted from the rest of the number. If the difference is 0 or a multiple of 7, then we say that the given number is divisible by 7. If we are not sure whether the resulting number is divisible by 7 or not, we repeat the same process with the resultant number. For example, in the number 154, let us multiply the last digit 4 by 2, which is 4 × 2 = 8. On subtracting 8 from 15, we get 7. 7 is divisible by 7 as it is the first multiple. Therefore, 154 is divisible by 7.
Using the Divisibility Rule of 7, Check if 145 is Divisible by 7.
By the divisibility rule of 7, the last digit should be multiplied by 2 and then subtracted with the rest of the number leaving the last digit. If the difference is 0 or a multiple of 7, then it is divisible by 7. For the given number 145, when we multiply the last digit 5 by 2, we get, 5 × 2 = 10. Now, on subtracting 10 from 14, we get 4. Since 4 is not a multiple of 7, we can conclude that 145 is not divisible by 7.
What is the Divisibility Rule of 7 and 11?
The divisibility rule of 7 tells us to pick the last digit of a number, multiply it by 2, and subtract the product from the rest of the number to its left. If the difference is 0 or a multiple of 7, then the given number is divisible by 7. According to the divisibility rule of 11, a number is divisible by 11 if the difference of the sum of the digits at the odd positions and even positions are either equal to 0 or a multiple of 11. In other words, the difference should be 0 or a number that 11 divides completely without leaving a remainder.
How do you know if a Large Number is Divisible by 7?
To know if a large number is divisible by 7 or not, we need to check the following conditions:
 Step 1: Pick the last digit of the large number.
 Step 2: Multiply it by 2. Subtract the product with the rest of the digits to its left leaving behind the last digit.
 Step 3: If the difference is 0 or a multiple of 7, then the number is divisible by 7.
 Step 4: If the difference is still a large number and we are not sure of its divisibility by 7, repeat the same steps from 1 to 3 with the number obtained in step 2.
How Many Numbers are there Between 1 and 100 which are Exactly Divisible by 7?
There are 14 numbers between 1 and 100 that are exactly divisible by 7. They are, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, and 98.
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