A number is divisible by 9 if the sum of its digits is divisible by 9.
Here's a breakdown of why this works and how to use it:
The Divisibility Rule Explained
The divisibility rule for 9 is based on modular arithmetic. Essentially, any power of 10 leaves a remainder of 1 when divided by 9 (e.g., 10 % 9 = 1, 100 % 9 = 1, 1000 % 9 = 1, and so on). Therefore, a number like 729 can be thought of as (7 100) + (2 10) + (9 1). When dividing by 9, we're essentially looking at the remainders, which simplifies to (7 1) + (2 1) + (9 1) = 7 + 2 + 9. If that sum is divisible by 9, the original number is also divisible by 9.
Steps to Check Divisibility by 9:
- Sum the Digits: Add all the digits of the number together.
- Check the Sum: Determine if the sum you calculated in step 1 is divisible by 9.
- Conclusion: If the sum is divisible by 9, then the original number is also divisible by 9. If the sum is not divisible by 9, then neither is the original number.
Examples:
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Example 1: 729
- Sum of digits: 7 + 2 + 9 = 18
- Is 18 divisible by 9? Yes (18 / 9 = 2)
- Therefore, 729 is divisible by 9.
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Example 2: 1233
- Sum of digits: 1 + 2 + 3 + 3 = 9
- Is 9 divisible by 9? Yes (9 / 9 = 1)
- Therefore, 1233 is divisible by 9.
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Example 3: 4567
- Sum of digits: 4 + 5 + 6 + 7 = 22
- Is 22 divisible by 9? No (22 / 9 = 2 with a remainder of 4)
- Therefore, 4567 is not divisible by 9.
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Example 4: A Larger Number: 657843
- Sum of digits: 6 + 5 + 7 + 8 + 4 + 3 = 33
- Is 33 divisible by 9? No (33 / 9 = 3 with a remainder of 6)
- Therefore, 657843 is not divisible by 9.
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Example 5: 725265
- Sum of digits: 7 + 2 + 5 + 2 + 6 + 5 = 27
- Is 27 divisible by 9? Yes (27 / 9 = 3)
- Therefore, 725265 is divisible by 9.
Summary:
The divisibility rule for 9 provides a quick and easy way to determine if a number is divisible by 9 without performing long division. Simply add up the digits, and if the resulting sum is divisible by 9, so is the original number.
