A number is divisible by 3 and 9 based on the sum of its digits.
Divisibility Rules Explained
The divisibility rules for 3 and 9 are based on a simple mathematical principle:
- Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Step-by-step Guide
- Calculate the Sum of Digits: Add all the individual digits of the number together.
- Divisibility by 3 Check: Divide the sum from step 1 by 3. If the result is a whole number, then the original number is divisible by 3.
- Divisibility by 9 Check: Divide the sum from step 1 by 9. If the result is a whole number, then the original number is divisible by 9.
Examples
Let's consider the number 657. Based on the provided reference, 657 is also divisible by 9:
- Calculate sum of digits: 6 + 5 + 7 = 18
- Divisibility by 3 Check: 18 / 3 = 6, which is a whole number. So, 657 is divisible by 3.
- Divisibility by 9 Check: 18 / 9 = 2, which is a whole number. So, 657 is divisible by 9 as well.
Let's try another number like 123:
- Calculate sum of digits: 1 + 2 + 3 = 6
- Divisibility by 3 Check: 6 / 3 = 2, which is a whole number. Therefore, 123 is divisible by 3.
- Divisibility by 9 Check: 6 / 9 = 0.666..., which is not a whole number. Thus, 123 is not divisible by 9.
Summary
| Divisibility Rule | Condition | Example |
|---|---|---|
| Divisible by 3 | The sum of the digits is divisible by 3. | 123, 657, 999 |
| Divisible by 9 | The sum of the digits is divisible by 9. | 657, 999, 189 |
Practical Insights
- A number divisible by 9 is always divisible by 3, but a number divisible by 3 is not necessarily divisible by 9.
- Understanding these rules can quickly determine divisibility without resorting to long division.
Conclusion
To determine if a number is divisible by 3 and 9, calculate the sum of its digits. If that sum is divisible by 3, the number is divisible by 3. If that sum is divisible by 9, the number is divisible by 9. The video in the reference explains this principle, stating that if 657 is divisible by 9, it means it can be broken into groups of 9.
