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zTests of Divisibility

Divisibility by 10 :

Consider multiples of 10. They are 10, 20, 30, 40...

Non-multiples of 10. They are 23, 31, 42, 55, 69... If the ones digit of a number is 0 then the number is a multiple of 10. If the ones digit is not 0 then the number is not a multiple of 10. Take the number
 



p is the one's digit, q is the ten's digit and r is the hundred's digit.

Since 10,100.... are divisible by 10, so are 10q, 100p... As for the number r is concerned, it must divisible by 10 if the given number is divisible by 10. This is possible only when
 

 

Divisibility by 5 :

The multiples of 5 are 5, 10, 15, 20, 25...

In one's place the digit is either 5 or 0.

If the ones digit of a number is 0 or 5 then it is divisible by 5.

Let us explain this rule. Any number ... pqr can be written as:
 

... + 100p + 10q + r

Since 10, 100 are divisible by 10 so are 10q, 100p, ... which in turn, are divisible

by 5 because 10 = 2 × 5. As far as number r is concerned it must be divisible by 5 if the number is divisible by 5. So r has to be either 0 or 5.
 

Divisibility by 2 :

The even numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22,...

The odd numbers are 3, 5, 7, 9, 11, 13, 15, 17, 19, 21...

The natural number is even if its one's digit is 2, 4, 6, 8 or 0.

A number is odd if its one's digit is 1, 3, 5, 7 or 9.

If the ones' digit of a number is 0, 2, 4, 6 or 8 then it is divisible by 2.

Any number written as .

are divisible by 2 because 100 and 10 are divisible by 2. As far as number r is concerned it must be divisible by 2 if the number is divisible by 2. So r has to be 0, 2, 4, 6 or 8 only then the number is divisible by 2.
 

Divisibility by 9 and 3.

In the divisibility by 10, 5 and 2 we use only the one's digit of the given number. We do not bother about rest of the digits. The divisibility is decided by the one's digit.

For checking divisibility by 9, let us consider 2689
 


 

If 9 + 6 + 8 + 2 is divisible by 9 or 3 then 2689 is divisible by 9 or 3

(i) A number N is divisible by 9 if the sum of its digits is divisible by 9. Otherwise it is not divisible by 9.

(ii) A number N is divisible by 3 if the sum of its digits is divisible by 3, otherwise it is not divisible by 3

Consider the number pqr
 

Divisibility by 9 (or 3) is possible if is divisible by 9 (or 3)
 

Example :

Check the divisibility of 12345678 by 9.

Solution
Solution :

The sum of the digits is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.

This number is divisible by 9.

 12345678 is divisible by 9
 


 


 

Example

Check the divisibility of 285751 by 9.

Solution

The sum of digits 2 + 8 + 5 + 7 + 5 + 1 = 28.

28 is not divisible by 9.

This number is not divisible by 9.


 

Example :

If the three digit number is divisible by 9. What is the value of

Solution :

Since is divisible by 9, sum of its digits should be divisible by 9. That is, should be divisible by 9.

This is possible only when


Example :

Check th divisibility of 4658217 by 3.

Solution :

The sum of the digits 4 + 6 + 5 + 8 + 2 + 1 + 7 = 33

33 is divisible by 3.

is divisible by 3.


 

Example :

Check the divisibility of 28517 by 3.

Solution :

The sum of the digits 2 + 8 + 5 + 1 + 7 = 23.

23 is not divisible by 3.

28517 is not divisible by 3.

 
 
 




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