How to Check Divisibility by 3 Using Digit Sum Rule with Examples
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 9 404 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 Explained with Simple Steps
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.
To check divisibility by 3:
- Add all the digits of the number.
- If the total is divisible by 3, then the original number is also divisible by 3.
2. How do you check if a number is divisible by 3?
You check if a number is divisible by 3 by adding its digits and seeing whether the sum is divisible by 3.
Steps:
- Add all the digits of the number.
- Check if the sum can be divided by 3 without remainder.
3. Why does the divisibility rule of 3 work?
The divisibility rule of 3 works because powers of 10 leave a remainder of 1 when divided by 3.
Any number like 456 can be written as:
- 4 × 100 + 5 × 10 + 6
4. What are some examples of numbers divisible by 3?
Numbers are divisible by 3 if their digit sum is divisible by 3.
Examples:
- 12 → 1 + 2 = 3 ✅
- 27 → 2 + 7 = 9 ✅
- 345 → 3 + 4 + 5 = 12 ✅
- 1002 → 1 + 0 + 0 + 2 = 3 ✅
5. Is 0 divisible by 3?
Yes, 0 is divisible by 3 because 0 divided by 3 equals 0 with no remainder.
Mathematically:
- 0 ÷ 3 = 0
6. What is the difference between the divisibility rule of 3 and 9?
The difference is that divisibility by 3 requires the digit sum to be divisible by 3, while divisibility by 9 requires the digit sum to be divisible by 9.
For example:
- 18 → 1 + 8 = 9 → divisible by 3 and 9
- 12 → 1 + 2 = 3 → divisible by 3 but not by 9
7. Can a number be divisible by both 2 and 3?
Yes, a number is divisible by both 2 and 3 if it is divisible by 6.
A number must:
- End in 0, 2, 4, 6, or 8 (divisible by 2).
- Have a digit sum divisible by 3.
8. How do you find the smallest number to add to make a number divisible by 3?
To find the smallest number to add, calculate the remainder when the digit sum is divided by 3 and subtract it from 3.
Steps:
- Find the digit sum.
- Divide the sum by 3 and find the remainder.
- If remainder is 1, add 2; if remainder is 2, add 1; if remainder is 0, add 0.
9. Does the divisibility rule of 3 work for large numbers?
Yes, the divisibility rule of 3 works for large numbers because it depends only on the sum of digits.
Example: 987654 → 9 + 8 + 7 + 6 + 5 + 4 = 39, and 39 ÷ 3 = 13, so 987654 is divisible by 3. The size of the number does not affect the rule.
10. What are common mistakes when using the divisibility rule of 3?
A common mistake is checking only the last digit instead of the sum of all digits.
Other mistakes include:
- Adding digits incorrectly.
- Forgetting to check if the digit sum is divisible by 3.
- Confusing the rule of 3 with the rule of 9.
