How to Check if a Number Is Divisible by 9 with Rule and Examples
What if I tell you that you can find if a number as big as 34117326 is divisible by 9 without doing the tedious process of long division? You might be thinking that it is some sort of magic trick when in reality it is one of the amazingly logical yet simple techniques of mathematics. If you’ve guessed it already, you’re right! We’re talking about the “Test of Divisibility of 9” Let’s get started on how this works.
Divisibility Test for 9
To test the divisibility of 9, we need to know its divisibility rule. Divisibility rules in Mathematics are a set of rules specific to each number which help us check whether that number is divisible by a particular number or not, without performing long division. Some common divisibility tests are for numbers 2 to 20.
Important Things to Recall
Good old addition (if you’re telling me 2+3 = 7, you first need to work on that).
Multiplication table of 9 or multiple of 9 (once you know 9 times 10 is 90, you are good to go!)
Multiplication Table of 9
Rule of Divisibility for 9
The rule of divisibility for 9 basically goes like this: “A number is divisible by 9 if the sum of all the digits present in the number is divisible by 9”.
To break down this complex-sounding rule, all you have to do is add up all the digits present in the number and see if that sum is divisible by 9.
Now, let’s check if this trick works with an example.
Let’s try it with 198
First, add all the digits in this number.
1 + 9 + 8 = 18
Now all you have to do to check is if 18 is divisible by 9, and the answer to that is, yes!
Hence, 198 is divisible by 9 too!
To get a better hang of this rule of divisibility, let’s practice with a few examples
Solved Examples
Q1. Check if 171 is divisible by 9.
Ans: 1+7+1=Z
9 is divisible by 9
Hence, 171 is divisible by 9
Q2. Check if 786 is divisible by 9.
Ans: 7+8+6=21
21 is not divisible by 9
Hence 786 is NOT divisible by 9
Q3. Check if 34117326 is divisible by 9.
Ans: 3+4+1+1+7+3+2+6=27
27 is divisible by 9
Hence 34117326 is also divisible by 9
“Practice makes a man perfect”. Now, let’s solve some more numbers to understand and master divisibility by 9.
Practice Problems
Is 725 divisible by 9? (Ans: Yes)
Is 5670 divisible by 9? (Ans: No)
Is 19997 divisible by 9? (Ans: Yes)
Summary
Let’s briefly summarise what we’ve learned up until now.
Divisibility by 9 can be found by adding the digits present in the number.
If the sum obtained is divisible by 9 (multiple of 9) the number is divisible by 9.
If the sum obtained is not divisible by (not a multiple of 9) the number is not divisible by 9.
FAQs on Divisible by 9 Rule Explained with Simple Steps
1. What is the rule for divisibility by 9?
A number is divisible by 9 if the sum of its digits is divisible by 9.
- Add all the digits of the number.
- If the total is 9, 18, 27, 36, etc., then the number is divisible by 9.
- If needed, repeat the process for large sums.
2. How do you check if a number is divisible by 9?
To check divisibility by 9, add all the digits and see if the sum is a multiple of 9.
- Example: 4,563 → 4 + 5 + 6 + 3 = 18.
- Since 18 is divisible by 9, 4,563 is also divisible by 9.
3. Why does the divisibility rule of 9 work?
The divisibility rule of 9 works because every power of 10 leaves a remainder of 1 when divided by 9. In base 10, 10 ≡ 1 (mod 9), so each digit contributes its face value to the remainder. Therefore, a number and the sum of its digits have the same remainder when divided by 9, which explains why the rule works mathematically.
4. What are some examples of numbers divisible by 9?
Some common numbers divisible by 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.
- Example: 81 → 8 + 1 = 9 (divisible by 9).
- Example: 1,458 → 1 + 4 + 5 + 8 = 18 (divisible by 9).
5. What is the difference between divisibility by 3 and divisibility by 9?
Both rules use the sum of digits, but divisibility by 9 is stricter than divisibility by 3.
- If the sum of digits is divisible by 3, the number is divisible by 3.
- If the sum of digits is divisible by 9, the number is divisible by 9.
6. Can you give a step-by-step example of testing divisibility by 9?
Yes, to test divisibility by 9, add the digits and check if the result is a multiple of 9.
- Example: 5,832
- Step 1: 5 + 8 + 3 + 2 = 18
- Step 2: 18 ÷ 9 = 2
7. Is 0 divisible by 9?
Yes, 0 is divisible by 9 because 0 divided by 9 equals 0 with no remainder. Since 0 = 9 × 0, it satisfies the definition of divisibility. Also, the sum of digits of 0 is 0, which is divisible by 9.
8. How do you find the smallest number to add to make a number divisible by 9?
To make a number divisible by 9, find the remainder when its digit sum is divided by 9 and subtract it from 9.
- Example: 234 → 2 + 3 + 4 = 9 (already divisible).
- Example: 235 → 2 + 3 + 5 = 10.
- 10 ÷ 9 leaves remainder 1, so add 8.
9. Can a negative number be divisible by 9?
Yes, negative numbers can be divisible by 9 if their absolute value is divisible by 9. For example, −81 is divisible by 9 because 81 ÷ 9 = 9. Divisibility rules apply the same way to positive and negative integers.
10. What is the formula for checking divisibility by 9?
The formula for divisibility by 9 is: a number is divisible by 9 if (sum of its digits) mod 9 = 0. In mathematical terms, if N is a number and S is the sum of its digits, then N is divisible by 9 when S ≡ 0 (mod 9). This modular arithmetic form explains the divisibility test clearly.
