Example:

Without actual division, check whether 654318 is divisible by 7.

 

Solution:

Here, we need to use the divisibility rule of 7.

 

To check the divisibility of a number by any prime number (other than 2), at first, we have to find a ‘key number’ for the prime number.

 

How to find key number ?

To find the ‘key number’ of any prime number,

1)           Multiply it by a number (the least number) such that the unit digit of the product will be 9.

2)         Then, add 1 to the product. Now, the sum becomes a multiple of 10.

3)         Divide the sum by 10. The quotient will be the ‘key number’.


So, we can find the key number of 7 as follows:




  


Thus, the key number for 7 is 5.


How to check divisibility ?

We can use the ‘key number’ to check the divisibility     as shown below:


(Extract the last digit and   multiply it by the key number and then add the product)










14 is divisible by 7.

Hence, the given number is divisible by 7.

 

Note: We can continue the process till we get a small number.

 

ALTERNATIVE METHOD:

We know that the key number for 7 is 5. Right ?

Let’s say that 5 is the first key number for 7.

Then, the second key number for 7 

           = 7 – the first key number

           = 7 – 5 = 2

 

How to check divisibility by using the 2nd key number ?

We can use the ‘2nd key number’ to check the divisibility     as shown below:


(Extract the last digit and   multiply it by the key number and then subtract the product)








63 is divisible by 7.

Hence, the given number is divisible by 7.

 

Note: We can continue the process till we get a small number.

 

PROBLEMS FOR PRACTICE:

1) By using the divisibility rule, check whether the following numbers are divisible by 7. (use 5 as the key number)

(a) 184744    

(b) 485273

 

2) By using the divisibility rule, check whether the following numbers are divisible by 7. (use 2 as the key number)

(a) 818205    

(b) 251979