An integer is divisible by 9 if the sum of its digits is divisible by 9.
Here's a more detailed explanation:
When a number is divisible by 9, it means that 9 divides into the number evenly, leaving no remainder. The divisibility rule for 9 provides a quick and easy way to determine if a number is divisible by 9 without performing long division.
Explanation of the Divisibility Rule for 9
The rule hinges on the principle that any power of 10 leaves a remainder of 1 when divided by 9. For example:
- 10 ÷ 9 = 1 remainder 1
- 100 ÷ 9 = 11 remainder 1
- 1000 ÷ 9 = 111 remainder 1
Consider the number 459. We can express it as:
(4 100) + (5 10) + (9 * 1)
Which can also be written as:
(4 (99 + 1)) + (5 (9 + 1)) + (9 * 1)
Expanding this, we get:
(4 99) + 4 + (5 9) + 5 + 9
Notice that (4 99) and (5 9) are both divisible by 9. Therefore, whether 459 is divisible by 9 depends only on whether (4 + 5 + 9) is divisible by 9. Since 4 + 5 + 9 = 18, and 18 is divisible by 9, then 459 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: If the sum of the digits is divisible by 9, then the original number is divisible by 9. If the sum is a multi-digit number, you can repeat step 1 until you get a single-digit number. That single-digit number must be 9 for the original number to be divisible by 9.
Examples:
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Example 1: Is 729 divisible by 9?
- 7 + 2 + 9 = 18
- 1 + 8 = 9
- Since 9 is divisible by 9, 729 is divisible by 9.
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Example 2: Is 1234 divisible by 9?
- 1 + 2 + 3 + 4 = 10
- 1 + 0 = 1
- Since 1 is not divisible by 9, 1234 is not divisible by 9.
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Example 3: Is 9999 divisible by 9?
- 9 + 9 + 9 + 9 = 36
- 3 + 6 = 9
- Since 9 is divisible by 9, 9999 is divisible by 9.
In Summary:
To quickly determine if an integer is divisible by 9, sum its digits. If the resulting sum is divisible by 9, then the original integer is also divisible by 9.
