Divisibility Rules for 3 and 9
The divisibility rules for 3 and 9 are based on the sum of digits of a number. Unlike the rules for 2, 5, and 10 (which check only the last digit), these rules require adding all the digits together.
These rules are powerful tools for mental maths and for checking whether long division will give a remainder or not.
What is Divisibility Rules for 3 and 9 - Class 4 Maths (Division (Grade 4))?
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.
| Divisible by | Rule | Example |
|---|---|---|
| 3 | Sum of digits divisible by 3 | 123: 1+2+3=6 (6÷3=2) → Yes |
| 9 | Sum of digits divisible by 9 | 729: 7+2+9=18 (18÷9=2) → Yes |
Divisibility Rules for 3 and 9 Formula
Divisible by 3 → Sum of all digits is divisible by 3
Divisible by 9 → Sum of all digits is divisible by 9
Key connection: Every number divisible by 9 is also divisible by 3 (because 9 is a multiple of 3). But not every number divisible by 3 is divisible by 9.
Solved Examples
Example 1: Example 1: Checking Divisibility by 3
Problem: Is 246 divisible by 3?
Solution:
Sum of digits = 2 + 4 + 6 = 12
Is 12 divisible by 3? 12 ÷ 3 = 4 (exact). Yes.
Answer: Yes, 246 is divisible by 3.
Example 2: Example 2: Not Divisible by 3
Problem: Is 325 divisible by 3?
Solution:
Sum of digits = 3 + 2 + 5 = 10
Is 10 divisible by 3? 10 ÷ 3 = 3 remainder 1. No.
Answer: No, 325 is not divisible by 3.
Example 3: Example 3: Checking Divisibility by 9
Problem: Is 5,634 divisible by 9?
Solution:
Sum of digits = 5 + 6 + 3 + 4 = 18
Is 18 divisible by 9? 18 ÷ 9 = 2. Yes.
Answer: Yes, 5,634 is divisible by 9.
Example 4: Example 4: Divisible by 3 but Not by 9
Problem: Is 123 divisible by 3? Is it divisible by 9?
Solution:
Sum of digits = 1 + 2 + 3 = 6
6 ÷ 3 = 2 (exact) → Divisible by 3. Yes.
6 ÷ 9 → 9 does not divide 6 exactly → Not divisible by 9. No.
Answer: 123 is divisible by 3 but not by 9.
Example 5: Example 5: Testing a Large Number
Problem: Is 8,127 divisible by 3? By 9?
Solution:
Sum of digits = 8 + 1 + 2 + 7 = 18
18 ÷ 3 = 6 → Divisible by 3. Yes.
18 ÷ 9 = 2 → Divisible by 9. Yes.
Answer: 8,127 is divisible by both 3 and 9.
Example 6: Example 6: Making a Number Divisible by 3
Problem: What digit should replace □ in 4□7 so the number is divisible by 3?
Solution:
Sum = 4 + □ + 7 = 11 + □
For divisibility by 3, (11 + □) must be divisible by 3.
Try values: 11+1=12 ✓, 11+4=15 ✓, 11+7=18 ✓
Answer: □ = 1, 4, or 7
Example 7: Example 7: Making a Number Divisible by 9
Problem: What digit should replace □ in 3□6 so the number is divisible by 9?
Solution:
Sum = 3 + □ + 6 = 9 + □
For divisibility by 9, (9 + □) must be divisible by 9.
9 + 0 = 9 ✓ and 9 + 9 = 18 ✓
Answer: □ = 0 or 9
Example 8: Example 8: Sorting Numbers
Problem: From the numbers 111, 234, 459, 612, 800, find which are divisible by 3 and which by 9.
Solution:
| Number | Digit Sum | ÷ 3? | ÷ 9? |
|---|---|---|---|
| 111 | 3 | Yes | No |
| 234 | 9 | Yes | Yes |
| 459 | 18 | Yes | Yes |
| 612 | 9 | Yes | Yes |
| 800 | 8 | No | No |
Example 9: Example 9: Word Problem
Problem: Meera has 738 beads. Can she divide them equally into 3 bags with none left over?
Solution:
Check if 738 is divisible by 3.
Sum of digits = 7 + 3 + 8 = 18. Since 18 ÷ 3 = 6, yes.
738 ÷ 3 = 246. Each bag gets 246 beads.
Answer: Yes, she can divide them equally.
Example 10: Example 10: Repeated Digit Sum
Problem: Is 9,999 divisible by 9?
Solution:
Sum of digits = 9 + 9 + 9 + 9 = 36
36 ÷ 9 = 4. Yes.
(You could also sum again: 3 + 6 = 9, which is divisible by 9.)
Answer: Yes, 9,999 is divisible by 9. (9,999 ÷ 9 = 1,111)
Real-World Applications
Divisibility rules for 3 and 9 are used in:
- Checking division: Quickly verify if a number divides by 3 or 9 without long division.
- Simplifying fractions: If both numerator and denominator have digit sums divisible by 3, the fraction can be simplified.
- Number puzzles: Finding missing digits in numbers.
- Error detection: In accounting, checking whether totals are divisible by 9 (casting out nines).
Key Points to Remember
- Divisibility by 3: Add all digits. If the sum is divisible by 3, the number is divisible by 3.
- Divisibility by 9: Add all digits. If the sum is divisible by 9, the number is divisible by 9.
- Every number divisible by 9 is also divisible by 3.
- Not every number divisible by 3 is divisible by 9 (e.g., 12 is divisible by 3 but not by 9).
- If the digit sum is large, you can add the digits again until you get a single digit.
- These rules work for numbers of any size.
Practice Problems
- Is 567 divisible by 3? By 9?
- Check if 4,851 is divisible by 9.
- Find the digit sum of 2,346. Is it divisible by 3?
- What digit can replace □ in 8□5 to make it divisible by 3?
- What digit can replace □ in 2□7 to make it divisible by 9?
- Dev has 891 stamps. Can he divide them into 9 equal groups?
- From the numbers 144, 253, 378, 410, 513, list those divisible by 3.
- Is 1,000 divisible by 3? By 9? Explain.
Frequently Asked Questions
Q1. Why does the digit sum test work for 3 and 9?
Because 10 leaves remainder 1 when divided by 9 (and by 3). So 100 leaves remainder 1, 1000 leaves remainder 1, and so on. This means a number has the same remainder as its digit sum when divided by 3 or 9.
Q2. If a number is divisible by 9, is it always divisible by 3?
Yes. Since 9 = 3 × 3, every multiple of 9 is also a multiple of 3. For example, 27 is divisible by both 9 and 3.
Q3. Can a number be divisible by 3 but not by 9?
Yes. For example, 15 has digit sum 6. Since 6 ÷ 3 = 2 (exact), it is divisible by 3. But 6 ÷ 9 has a remainder, so 15 is not divisible by 9.
Q4. What if the digit sum itself is a large number?
Keep adding digits until you get a small number. For example, 8,991: digit sum = 8+9+9+1 = 27. Still large? 2+7 = 9. Since 9 is divisible by both 3 and 9, so is 8,991.
Q5. How is the rule for 3 different from the rule for 9?
Both use the digit sum, but the sum must be divisible by 3 for the rule of 3, and by 9 for the rule of 9. The rule of 9 is stricter — fewer numbers satisfy it.
Q6. Can these rules be applied to 5-digit or 6-digit numbers?
Yes. The digit sum rule works for any number, no matter how large. Just add all the digits.
Q7. How do I find a missing digit using these rules?
Set up: known digits sum + □ must be divisible by 3 (or 9). Try digits 0 through 9 and see which values make the total sum divisible.
Q8. Are divisibility rules for 3 and 9 in the NCERT Class 4 syllabus?
NCERT Class 4 introduces patterns related to multiples of 3 and 9. The formal digit sum rule is emphasised more in Class 5, but many CBSE schools introduce it in Class 4 as well.
