Divisibility rules help us quickly determine if a number can be divided evenly by another without performing long division. One useful rule is the divisibility rule for 9.
More Examples of Divisibility by 9 Check
729 : Sum the digits = 7 + 2 + 9 = 18 which is divisible by 9. So, 729 is divisible by 9.
5463 : Sum the digits = 5 + 4 + 6 + 3 = 18 which is divisible by 9. So, 5463 is divisible by 9.
Proof of the Divisibility by 9 Rule
Please note that a number would be divisible by 9 if the remainder is 0 when divided by 9.
The main reason for the rule's working is, that when we divide any power of 10 with 9, we always get the remainder as 1. For example 100, 101, 102, 103, ... all give remainder as 1 when divided by 9.
For any number say 5463, we can re-write it as
5 x 103 + 4 x 102 + 6 x 101 + 3
When we find remainder with the above expression, we can notice that all powers of 10 would give remainder 1, so the remainder is going to be same as remainder we get from 5 + 4 + 6 + 3 which is sum of digits of 9. So the remainder of a number when divided by 9 is same as the remainder when its digit sum is divided.
What is the Remainder when Divided by 9?
As we can see from the proof, the above rule can be used to find remainder also. When we divide a large number by 9, the remainder is same when sum of its digits is divided by 9. For example, the remainder 12121 when divided by 9 is same as the remainder of (! + 2 + 1 + 2 + 1) when divided by 9 which is 7
Verification with Table of 9
The following are the numbers in table of 9 and their sum of digits. We can clearly see that all digit sums are multiples of 9.
9 : 9
18 : 1 + 8
27 : 2 + 7
36 : 3 + 6
45 : 4 + 5
54 : 5 + 4
63 : 6 + 3
72 : 7 + 2
81 : 8 + 1
90 : 9 + 0
Other Divisibility Rules:
Solved Examples on Divisibility by 9
Example 1: Is 9738 divisible by 9?
Solution:
Given number is 9738
To check divisibility by 9 we will sum individual digit of given numbers
Add the digits: 9 + 7 + 3 + 8 = 27
Check the sum: 27 is divisible by 9
So, 9738 is divisible by 9.
Example 2: Check if 2765 divisible by 9?
Solution:
Given number is 2765
To check divisibility by 9 we will sum individual digit of given numbers
Add the digits: 2 + 7 + 6 + 5 = 20
Check the sum: 20 not is divisible by 9
So, 2765 is not divisible by 9.
Example 3: Find the smallest 2 digit number divisible by 9?
Solution:
The smallest 2-digit number is 10
When we divide 10 by 9 then remainder 1 left
If we subtract 9 from remainder then 8 left
Now, we have to add 8 to 10 to get the smallest 2-digit number which is completely divisible by 9
When we add = 10 + 8 = 18
Now smallest 2-digit divisible by 9 is 18.
Example 4: How many numbers between 400 and 500 are divisible by 9?
Solution:
First, find the smallest number divisible by 9 that is greater than or equal to 400
Smallest number between 400 and 500 that is divisible by 9 is 405
Now, find the largest number divisible by 9 that is less than or equal to 500
Largest number between 400 and 500 that is divisible by 9 is 495
Now total numbers between 405 and 495 ,
Using nth term formula of A.P. where first term is 405, common difference is 9 and nth term is 495
495 = 405 + (n − 1) × 9
⇒ 495 = 405 + 9n − 9
⇒ 495 =396 + 9n
⇒ 495 − 396 = 9n
⇒ 99 = 9n
⇒ n = 99 / 9 = 11.
Therefore, there are 11 numbers between 400 and 500 that are divisible by 9.
Practice Questions on Divisibility by 9
Problem 1: Is the number 486 divisible by 9?
Problem 2: Determine if 729 is divisible by 9.
Problem 3: Check if 1,125 is divisible by 9.
Problem 4: Is the number 3,618 divisible by 9?
Problem 5: Determine if 5,832 is divisible by 9.
Problem 6: Check if 14,553 is divisible by 9.
Problem 7: Is the number 27,891 divisible by 9?
Problem 8: Determine if 45,018 is divisible by 9.
Problem 9: Check if 67,527 is divisible by 9.
Problem 10: Is the number 81,099 divisible by 9?
Also Read: Practice Questions on Divisibility Rules
Conclusion
In conclusion, the rule of divisibility by 9 is a simple yet powerful tool that helps quickly determine if a number can be divided evenly by 9 without remainder. By simply adding the digits of a number and checking if the resulting sum is divisible by 9, we can easily verify the divisibility of any number.