If the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9. This is the divisibility rule for 9.
Explanation of the Divisibility Rule of 9
The divisibility rule for 9 is a handy shortcut for determining whether a number can be divided evenly by 9 without performing the actual division. It relies on the principle that any number can be expressed as a sum of its digits multiplied by powers of 10. Since any power of 10 is congruent to 1 modulo 9 (i.e., leaves a remainder of 1 when divided by 9), the remainder of a number when divided by 9 is the same as the remainder of the sum of its digits when divided by 9.
Examples
-
Example 1: Consider the number 54. The sum of its digits is 5 + 4 = 9. Since 9 is divisible by 9, the number 54 is also divisible by 9 (54 / 9 = 6).
-
Example 2: Consider the number 126. The sum of its digits is 1 + 2 + 6 = 9. Since 9 is divisible by 9, the number 126 is also divisible by 9 (126 / 9 = 14).
-
Example 3: Consider the number 981. The sum of its digits is 9 + 8 + 1 = 18. Since 18 is divisible by 9, the number 981 is also divisible by 9 (981 / 9 = 109).
-
Example 4 (Larger Number): Consider the number 41472. The sum of its digits is 4 + 1 + 4 + 7 + 2 = 18. Since 18 is divisible by 9, the number 41472 is also divisible by 9 (41472 / 9 = 4608).
How to Use the Rule
- Sum the digits: Add up all the digits of the number.
- Check divisibility by 9: Determine if the sum of the digits is divisible by 9.
- Conclusion: If the sum is divisible by 9, the original number is also divisible by 9. If the sum is not divisible by 9, the original number is not divisible by 9.
Why is This Useful?
The divisibility rule of 9 is particularly useful for:
- Quickly checking divisibility: It allows you to determine if a number is divisible by 9 without performing long division.
- Simplifying calculations: It can help simplify certain calculations by identifying multiples of 9.
- Error detection: It can be used to detect errors in arithmetic, especially when dealing with large numbers.
