Divisibility Rules for 2, 5 and 10
Divisibility rules are quick tests that tell us whether a number can be divided exactly by another number (remainder = 0) without actually performing the division.
In Class 4, students learn the divisibility rules for 2, 5, and 10 first, because they all depend on the last digit (ones digit) of the number, making them the easiest rules to apply.
What is Divisibility Rules for 2, 5 and 10 - Class 4 Maths (Division (Grade 4))?
A number is divisible by another number if the division gives remainder 0 (exact division).
| Divisible by | Rule | Examples (Yes) | Examples (No) |
|---|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 (even) | 24, 156, 3,780 | 31, 57, 243 |
| 5 | Last digit is 0 or 5 | 25, 140, 2,005 | 32, 78, 511 |
| 10 | Last digit is 0 | 30, 450, 7,000 | 25, 98, 345 |
Divisibility Rules for 2, 5 and 10 Formula
Divisible by 2 → Last digit is even (0, 2, 4, 6, 8)
Divisible by 5 → Last digit is 0 or 5
Divisible by 10 → Last digit is 0
Types and Properties
Observations about these rules:
- Every number divisible by 10 is also divisible by both 2 and 5 (since 10 = 2 × 5).
- Not every number divisible by 2 is divisible by 5. Example: 24 is divisible by 2 but not by 5.
- Not every number divisible by 5 is divisible by 2. Example: 35 is divisible by 5 but not by 2.
- Numbers divisible by both 2 and 5 must be divisible by 10.
Solved Examples
Example 1: Example 1: Checking Divisibility by 2
Problem: Is 4,386 divisible by 2?
Solution:
Last digit = 6 (even number)
Since the last digit is even, 4,386 is divisible by 2.
Answer: Yes, 4,386 is divisible by 2.
Example 2: Example 2: Checking Divisibility by 5
Problem: Is 2,473 divisible by 5?
Solution:
Last digit = 3
The last digit must be 0 or 5 for divisibility by 5. Since 3 is neither, 2,473 is not divisible by 5.
Answer: No, 2,473 is not divisible by 5.
Example 3: Example 3: Checking Divisibility by 10
Problem: Is 5,670 divisible by 10?
Solution:
Last digit = 0
Since the last digit is 0, the number is divisible by 10.
Answer: Yes, 5,670 is divisible by 10.
Example 4: Example 4: Testing All Three Rules
Problem: Check whether 450 is divisible by 2, 5, and 10.
Solution:
Last digit = 0
- Divisible by 2? Last digit is 0 (even). Yes.
- Divisible by 5? Last digit is 0. Yes.
- Divisible by 10? Last digit is 0. Yes.
Answer: 450 is divisible by 2, 5, and 10.
Example 5: Example 5: Sorting Numbers
Problem: From the numbers 125, 248, 360, 517, 695, sort them into: divisible by 2, by 5, by 10.
Solution:
| Number | Last Digit | ÷ 2? | ÷ 5? | ÷ 10? |
|---|---|---|---|---|
| 125 | 5 | No | Yes | No |
| 248 | 8 | Yes | No | No |
| 360 | 0 | Yes | Yes | Yes |
| 517 | 7 | No | No | No |
| 695 | 5 | No | Yes | No |
Example 6: Example 6: Word Problem (Sharing Evenly)
Problem: Ria has 346 stickers. Can she share them equally between 2 friends (without any left over)?
Solution:
Check if 346 is divisible by 2.
Last digit = 6 (even). Yes, 346 is divisible by 2.
346 ÷ 2 = 173. Each friend gets 173 stickers.
Answer: Yes, she can share them equally.
Example 7: Example 7: Making a Number Divisible
Problem: What digit should replace □ in 73□ so that the number is divisible by 5?
Solution:
For divisibility by 5, the last digit must be 0 or 5.
So □ = 0 or □ = 5.
The numbers are 730 and 735.
Answer: □ = 0 or 5
Example 8: Example 8: Even and Odd Connection
Problem: Why are all even numbers divisible by 2?
Solution:
Even numbers end in 0, 2, 4, 6, or 8. The divisibility rule for 2 says: a number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8.
So every even number satisfies this rule.
Answer: Even numbers have an even last digit, which is exactly the condition for divisibility by 2.
Example 9: Example 9: Largest 3-Digit Number Divisible by 10
Problem: Find the largest 3-digit number divisible by 10.
Solution:
The largest 3-digit number is 999. But 999 ends in 9, not 0.
The largest 3-digit number ending in 0 is 990.
Check: 990 ÷ 10 = 99 (exact). ✓
Answer: 990
Example 10: Example 10: Divisible by Both 2 and 5
Problem: Find all 2-digit numbers divisible by both 2 and 5.
Solution:
A number divisible by both 2 and 5 must be divisible by 10 (since 2 × 5 = 10).
Last digit must be 0.
2-digit numbers ending in 0: 10, 20, 30, 40, 50, 60, 70, 80, 90.
Answer: 10, 20, 30, 40, 50, 60, 70, 80, 90 (9 numbers).
Real-World Applications
Divisibility rules save time in many situations:
- Mental maths: Quickly check if a number divides evenly before doing long division.
- Sharing: Check if items can be shared equally among 2, 5, or 10 people.
- Money: ₹5 and ₹10 coins/notes — check if a total can be paid exactly.
- Finding factors: Divisibility rules help identify factors quickly.
Key Points to Remember
- Divisible by 2: Last digit is even (0, 2, 4, 6, 8).
- Divisible by 5: Last digit is 0 or 5.
- Divisible by 10: Last digit is 0.
- Every number divisible by 10 is automatically divisible by both 2 and 5.
- A number divisible by both 2 and 5 is divisible by 10.
- These rules depend only on the last digit, making them very fast to apply.
Practice Problems
- Is 7,894 divisible by 2? By 5? By 10?
- Check which of these numbers are divisible by 5: 135, 200, 478, 605, 711.
- Find all 3-digit numbers ending in 0 that are between 300 and 400.
- What digit can replace □ in 56□ to make the number divisible by 10?
- Dev has 275 marbles. Can he divide them into groups of 5 with none left over?
- Is 1,000 divisible by 2, 5, and 10? Explain.
- Find the smallest 4-digit number that is divisible by both 2 and 5.
- Priya says every number ending in 5 is divisible by both 5 and 10. Is she correct? Why or why not?
Frequently Asked Questions
Q1. Why do divisibility rules for 2, 5, and 10 depend only on the last digit?
Because 10 is divisible by 2, 5, and 10. Any number can be written as (some multiple of 10) + (last digit). The multiple of 10 part is always divisible by 2, 5, and 10, so only the last digit matters.
Q2. Is every even number divisible by 2?
Yes. Even numbers end in 0, 2, 4, 6, or 8 — exactly the digits that satisfy the divisibility rule for 2.
Q3. Can a number be divisible by 5 but not by 10?
Yes. Numbers ending in 5 (like 15, 25, 135) are divisible by 5 but not by 10. Only numbers ending in 0 are divisible by 10.
Q4. Is 0 divisible by 2, 5, and 10?
Yes. 0 ÷ 2 = 0, 0 ÷ 5 = 0, 0 ÷ 10 = 0. All give remainder 0. Zero is divisible by every non-zero number.
Q5. What is the difference between 'divisible by' and 'divides'?
They describe the same relationship from different sides. '12 is divisible by 3' means the same as '3 divides 12'. Both mean 12 ÷ 3 has remainder 0.
Q6. How are divisibility rules useful?
They let you check division without calculating. This is helpful for finding factors, simplifying fractions, and solving problems mentally.
Q7. If a number is divisible by both 2 and 5, must it be divisible by 10?
Yes. If the last digit satisfies both rules (even AND 0 or 5), the only digit that is both even and in {0, 5} is 0. So the number ends in 0, which means it is divisible by 10.
Q8. Are divisibility rules for 2, 5, and 10 in the NCERT Class 4 syllabus?
Yes. NCERT Class 4 Maths introduces divisibility concepts through patterns in multiplication tables and observations about even/odd numbers.
