Divisibility by 9 is a useful concept in arithmetic that helps us quickly determine if a number can be divided by 9 without leaving a remainder. The rule for checking divisibility by 9 is simple but powerful.
The Rule
A number is divisible by 9 if the sum of its digits is also divisible by 9. This means you add up all the digits in the number, and if that sum can be divided by 9 without a remainder, then the original number is divisible by 9.
Example
Let’s take the number 729 as an example:
- Add the digits: $7 + 2 + 9 = 18$
- Check if the sum (18) is divisible by 9: $18 text{ divided by } 9 = 2$ (no remainder)
Since 18 is divisible by 9, 729 is also divisible by 9.
Why Does This Work?
The rule works because of the properties of numbers in the base-10 system. When you break down a number into its individual digits, each digit represents a power of 10. For example, 729 can be written as:
$7 times 10^2 + 2 times 10^1 + 9 times 10^0$
When you add the digits together, you are essentially reducing the number modulo 9, which simplifies the process of checking for divisibility by 9.
Practice Problems
Here are a few practice problems to help you understand:
- Is 123 divisible by 9?
- Sum of digits: $1 + 2 + 3 = 6$
- 6 is not divisible by 9, so 123 is not divisible by 9.
- Is 567 divisible by 9?
- Sum of digits: $5 + 6 + 7 = 18$
- 18 is divisible by 9, so 567 is divisible by 9.
- Is 81 divisible by 9?
- Sum of digits: $8 + 1 = 9$
- 9 is divisible by 9, so 81 is divisible by 9.
Conclusion
Understanding the rule for divisibility by 9 is a handy tool for quickly determining whether a number can be divided by 9 without a remainder. By simply adding the digits of the number and checking if the sum is divisible by 9, you can save time and effort in your calculations.
3. Wikipedia – Divisibility Rule