How to Check if a Number is Divisible by 3: Step-by-Step Guide
The Divisibility Rule of 3 is an essential shortcut in number theory and arithmetic. This rule allows students to quickly check if numbers—small or very large—are divisible by 3, a skill especially important for saving time in maths exams, competitive entrance tests, and real-world calculations. By mastering this concept, students can boost their mental maths speed and confidence.
Understanding the Divisibility Rule of 3
The Divisibility Rule of 3 states: A whole number is divisible by 3 if the sum of its digits is a multiple of 3. If you add up all the digits of any number, and the result is a number that 3 divides evenly (no remainder), then the original number can also be divided exactly by 3.
For example, in the number 246: 2 + 4 + 6 = 12; since 12 is divisible by 3, so is 246.
Step-by-Step: How To Apply the Divisibility Rule of 3
- Write down the number you want to check.
- Add together all its digits.
- See if the sum is a multiple of 3 or is divisible by 3.
- If yes, the original number is divisible by 3. If not, it isn’t.
Worked Examples
| Number | Sum of Digits | Divisible by 3? | Reason |
|---|---|---|---|
| 645 | 6+4+5=15 | Yes | 15 ÷ 3 = 5, so 645 is divisible by 3 |
| 429714 | 4+2+9+7+1+4=27 | Yes | 27 ÷ 3 = 9, so 429714 is divisible by 3 |
| 398 | 3+9+8=20 | No | 20 is not divisible by 3 |
Divisibility Rule of 3 vs Other Numbers
| Number | Divisibility Rule |
|---|---|
| 2 | The last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of all digits is divisible by 3 |
| 5 | The last digit is 0 or 5 |
| 6 | Number is divisible by both 2 and 3 |
| 7 | Double the last digit and subtract from the rest, if result is 0 or multiple of 7 |
| 9 | Sum of all digits is divisible by 9 |
To learn more about related rules, check out our Divisibility Rules – Overview and detailed guides like the Divisibility Rule of 9 and Divisibility Rule of 2.
Practice Problems
- Is 1,045 divisible by 3?
- Check if 2,412 is divisible by 3.
- Among 999 and 997, which is divisible by 3?
- Is 100,203 divisible by 3?
- Is 591 divisible by 3 or 9 or both?
- Check if 12,345,678 is divisible by 3.
- Find if 200 is divisible by 3.
- Is 381 divisible by 3?
- Is 4,238 divisible by 3 and by 2?
- Which is the smallest 3-digit number divisible by 3?
Common Mistakes to Avoid
- Confusing the divisibility rule of 3 with that of 9—check what the rule asks for!
- Adding the digits incorrectly. Double check your sums.
- Forgetting that 0 counts as a digit—always include zeros in the sum.
- Thinking a negative number behaves differently. The rule works the same way for negative numbers!
- Trying the rule for decimals—it only works for whole numbers.
Real-World Applications
Learning the divisibility rule of 3 is useful far beyond school. It helps in:
- Mental maths and quick calculations during exams like JEE, NEET, and Olympiads
- Dividing items into equal groups in everyday life
- Banking and finance, for quick cheque and currency checks
- Coding and computer science, error-checking in barcodes or data validation
- Fun maths puzzles and games that use divisibility concepts
At Vedantu, we simplify number theory concepts like divisibility so you can apply them confidently in both exams and the real world.
Divisibility rules—especially the divisibility rule of 3—offer fast, reliable shortcuts for checking division without long calculations. By practising these rules and understanding related concepts like factors and multiples, students can improve both their calculation speed and accuracy. For deeper learning, you can also explore Prime Numbers, Number System, and Problems On Divisibility Rules.
In summary, the divisibility rule of 3 allows you to swiftly check divisibility, helping with mental maths, exam shortcuts, and real-world division problems. Mastering this rule will make maths easier and faster for everyday calculations and competitive tests alike.
FAQs on Divisibility Rule of 3 – Definition, Steps & Solved Examples
1. What is the divisibility rule of 3?
The divisibility rule of 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3. For example, 27 is divisible by 3 because 2 + 7 = 9, and 9 is divisible by 3.
2. How do you test if a number is divisible by 3?
To check if a number is divisible by 3, add all its digits together. If this sum is divisible by 3 (i.e., the sum is a multiple of 3), then the original number is also divisible by 3. Otherwise, it's not divisible by 3. This is a quick and easy divisibility test.
3. Is 429714 divisible by 3?
Let's check: 4 + 2 + 9 + 7 + 1 + 4 = 27. Since 27 is divisible by 3 (27/3 = 9), then 429714 is divisible by 3.
4. Is 645 divisible by 3?
Add the digits: 6 + 4 + 5 = 15. Since 15 is divisible by 3 (15/3 = 5), 645 is divisible by 3.
5. What are divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10?
Here's a summary of common divisibility rules:
- 2: The last digit is even (0, 2, 4, 6, 8).
- 3: The sum of the digits is divisible by 3.
- 4: The last two digits are divisible by 4.
- 5: The last digit is 0 or 5.
- 6: The number is divisible by both 2 and 3.
- 8: The last three digits are divisible by 8.
- 9: The sum of the digits is divisible by 9.
- 10: The last digit is 0.
6. What is the difference between divisibility rules for 3 and 9?
Both rules involve summing the digits. A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9. Therefore, any number divisible by 9 is also divisible by 3, but not vice versa.
7. What if the sum is a large number?
If the sum of the digits is a large number, you can repeatedly apply the rule until you get a small sum. For example, if the sum is 57, add 5+7=12. Then, add 1+2=3, Since 3 is divisible by 3, the original number is divisible by 3.
8. Can you use the rule for negative numbers?
Yes, the rule applies to negative numbers. Ignore the negative sign, sum the digits, and check if the sum is divisible by 3. If it is, the original negative number is divisible by 3.
9. How can I tell if 645 is divisible by 3?
Add the digits of 645: 6 + 4 + 5 = 15. Since 15 is divisible by 3, then 645 is divisible by 3.
10. What is the divisibility rule of 3 and 9?
A number is divisible by 3 if the sum of its digits is divisible by 3. A number is divisible by 9 if the sum of its digits is divisible by 9. This means that any number divisible by 9 is also divisible by 3, but the opposite isn't always true.
