Divisibility Rules for 2, 3, 4, 5, 6, 9, 10 & 11 Explained
Understanding Division by Special Numbers is a crucial arithmetic skill for students in classes 3–6 and beyond. It helps you solve division questions much faster, especially during school exams and competitive tests. Mastering this concept will improve your overall number sense, calculation speed, and confidence in maths.
What is Division by Special Numbers?
Division by special numbers means dividing numbers by specific, often-used divisors like 2, 3, 4, 5, 6, 9, 10, and 11. These numbers are “special” because they have easy-to-remember rules (called divisibility rules) that help you quickly check if a number can be divided by them, without a full long division calculation. These tricks are common in arithmetic, fractions, multiples, and algebra.
For example, if you know the rule for 4 (if the last two digits are divisible by 4, the whole number is), you can save time in both exams and daily maths tasks. You’ll also use these rules for simplifying fractions, checking factors, and finding greatest common factors (GCF/LCM)—see our guides on Divisibility Rules and Multiplication and Division of Integers for more!
Table: Divisibility Rules for Special Numbers
The table below gives you all the shortcut rules for quick divisibility checks. Remembering them helps you spot patterns and solve division problems easily:
| Divisor | Divisibility Rule | Quick Example |
|---|---|---|
| 2 | Number is even (last digit 0, 2, 4, 6, 8). | 48 (last digit 8) is divisible by 2. |
| 3 | Sum of digits is divisible by 3. | 132 (1+3+2=6, divisible by 3). |
| 4 | Last two digits form a number divisible by 4. | 316 (16 ÷ 4 = 4). |
| 5 | Last digit is 0 or 5. | 355 ends in 5. |
| 6 | Divisible by both 2 and 3. | 132 (even, 1+3+2=6). |
| 9 | Sum of digits is divisible by 9. | 459 (4+5+9=18). |
| 10 | Last digit is 0. | 260 ends in 0. |
| 11 | Difference of sum of digits in odd and even positions is 0 or 11. | 506 → (5+6) - 0 = 11. |
For a deeper dive, visit our page on Divisibility Rules.
Step-by-Step Division Examples
Let’s put these rules into action with practical, exam-ready problems! Follow these steps to master both mental division checks and written long division.
Example 1: Dividing by 4 using the shortcut
Is 732 divisible by 4?
- Check the last two digits: 32.
- 32 ÷ 4 = 8 (no remainder). Yes, 732 is divisible by 4.
Try the same logic with bigger numbers. Fast, right?
Example 2: Long Division by 39
Find 87652 ÷ 39.
- Estimate: 39 × 2 = 78, 39 × 20 = 780, etc.
- Set up long division:
- 39 goes into 87 two times (2×39=78), remainder 9.
- Bring down 6, now 96. 39 goes into 96 two times (2×39=78), remainder 18.
- Bring down 5, now 185. 39 goes into 185 four times (4×39=156), remainder 29.
- Bring down 2, now 292. 39 goes into 292 seven times (7×39=273), remainder 19.
So, 87652 ÷ 39 = 2247 remainder 19; written as 2247 R19 or 2247.19/39.
Example 3: Quick check for 11
Is 5064 divisible by 11?
- Sum digits in odd places: 5 + 6 = 11
- Sum digits in even places: 0 + 4 = 4
- Difference: 11 - 4 = 7 (not 0 or 11)
So, 5064 is not divisible by 11.
Want more? Practice with our Division guide for stepwise support!
Division Tricks and Shortcuts
- To divide by 5: Multiply the number by 2, then divide by 10.
Example: 48 ÷ 5 = (48 × 2) ÷ 10 = 96 ÷ 10 = 9.6 - By 9: If the digit sum is divisible by 9, so is the number.
Example: 24786 (2+4+7+8+6=27; 27÷9=3) - By 11: Alternate digit sum trick (explained above).
- Avoid dividing blindly—always check for patterns and apply rules if possible to save time!
Tip: For two-digit divisors (like 25, 50, 20), rewrite the division as multiplying by a decimal or fraction.
Example: 120 ÷ 25 = 120 × (1/25) = 120 × 0.04 = 4.8
Explore more math shortcuts at our Maths Tricks page!
Worksheets & Practice Problems
Ready to test your knowledge? Try these practice questions. Answers are given below for quick self-check!
- Which numbers below are divisible by 3? 219, 275, 672, 1428
- Is 360 divisible by 4? Show your step.
- Without dividing, tell if 785 is divisible by 5 and 10.
- Solve: 893 ÷ 11 (Use the divisibility rule first).
- Fill in the blank: A number ending in ___ is always divisible by 2.
Answers:
- 219 (2+1+9=12), 672 (6+7+2=15), 1428 (1+4+2+8=15) — all divisible by 3; 275 is not.
- 360 (last two digits 60 ÷ 4 = 15) — Yes, divisible.
- 785 ends in 5 (so divisible by 5), but not by 10 (should end in 0).
- 893: 8+3+9=20, not a multiple of 11; use long division (893 ÷ 11 = 81 R2).
- Any even digit (0, 2, 4, 6, 8).
Find more exercises and answers in our Division by Special Numbers Worksheets.
Real-World Applications
Division by special numbers is everywhere! When you split ₹200 among 4 friends, check if 300 pages can be grouped into packets of 6, or see if your clock shows a time divisible by 5, you’re applying these rules. Accountants, engineers, chefs, and even shopkeepers use divisibility shortcuts daily to make quick, accurate calculations. Understanding these helps you reason faster in all kinds of math and practical life situations!
Common Mistakes to Avoid
- Mixing up rules: Don’t confuse “sum of digits” (for 3 or 9) with “last two digits” (for 4).
- Forgetting to check both parts for 6 (must be divisible by both 2 and 3).
- Assuming every number ending with 5 is divisible by 10 (must end with 0 for 10).
- Stopping the check too early for the rule of 11 — finish all additions and subtractions first.
Tip: Always double-check your divisibility with actual calculation when in doubt!
Page Summary
In this topic, we learned how mastering Division by Special Numbers helps with faster, more accurate maths in school and in life. Using shortcut rules for divisibility saves exam time and builds deep mathematical understanding. With Vedantu resources and regular practice, you’ll ace all division challenges confidently!
This essential topic connects closely to Prime Numbers, fractions, and multiplication tables, so keep exploring to build even more skills!
FAQs on Division by Special Numbers: Easy Rules, Tricks & Practice
1. What is the quickest way to check if a number is divisible by 4?
To quickly check if a number is divisible by 4, examine its last two digits. If this two-digit number is divisible by 4, then the entire number is divisible by 4. For example, consider the number 1312. The last two digits are 12, which is divisible by 4 (12/4=3). Therefore, 1312 is divisible by 4.
2. How do you divide large numbers by special numbers using long division?
Long division works the same for large numbers as it does for smaller ones. Divide the leading digit(s) of the dividend by the divisor. Write the quotient above and bring down the next digit. Repeat this until all digits have been processed. Consider using divisibility rules to simplify before the long division. Practice is key!
3. What are the divisibility rules for 11?
The divisibility rule for 11 involves alternating sums and differences of digits. Starting from the rightmost digit, alternately add and subtract digits. If the result is divisible by 11, the number itself is divisible by 11. For example, let's consider the number 91827. We compute (7 - 2 + 8 - 1 + 9) = 21. Since 21 is not divisible by 11, neither is 91827. However, 121 works as (1-2+1)=0, which is divisible by 11.
4. Can these tricks be used for mental maths in exams?
Yes! Divisibility rules and shortcuts are excellent for mental math in exams. They save time and reduce errors. Mastering these techniques improves speed and accuracy for division calculations.
5. Where can I find division by special numbers worksheets with answers?
Vedantu provides numerous worksheets and practice problems on division. Search our website for 'division by special numbers worksheets' to find suitable resources for practice.
6. What is 87652 divided by 39 long division?
To solve 87652 divided by 39 using long division, you would perform the standard long division algorithm. The quotient is 2247 and the remainder is 9. Practice with more examples to master the technique.
7. What is the divisibility rule of 2, 3, 4, 5, 6, 7, 8, 9, 10?
Divisibility rules provide quick ways to check divisibility without performing the actual division. Here are some examples: A number is divisible by 2 if its last digit is even; by 3 if the sum of its digits is divisible by 3; by 5 if its last digit is 0 or 5, and so on. Refer to a divisibility rules chart for a complete list.
8. What is 7 divided by 777 long division?
When dividing 7 by 777 using long division, you'll find the quotient is 0 and the remainder is 7 (because 7 is smaller than 777). Remember that when the dividend is less than the divisor, the quotient is 0.
9. What is 67 division by?
This question is a bit ambiguous. 67 can be divided by many numbers (its divisors). To find them, you would typically determine the factors of 67. It's a prime number; its only factors are 1 and 67. That means the only whole numbers that divide 67 evenly are 1 and 67.
10. What is the difference between division and divisibility?
Division is the process of splitting a number (dividend) into equal parts by another number (divisor). Divisibility refers to whether a number can be divided by another number without leaving a remainder. For example, 12 divided by 3 is 4 (division). 12 is divisible by 3 (divisibility).
11. How to check if a number is divisible by 6?
A number is divisible by 6 if it is divisible by both 2 and 3. Check if the last digit is even (divisible by 2) and if the sum of its digits is divisible by 3. If both conditions are true, then the number is divisible by 6.
12. How to check if a number is divisible by 9?
A number is divisible by 9 if the sum of its digits is divisible by 9. For example, consider 81. The sum of its digits (8+1=9) is divisible by 9, therefore, 81 is divisible by 9.
13. How to check if a number is divisible by 10?
A number is divisible by 10 if its last digit is 0. This is a simple rule to remember. For example, 120 is divisible by 10 because its last digit is 0.
14. How to check if a number is divisible by 5?
A number is divisible by 5 if its last digit is either 0 or 5. For example, 15 and 20 are both divisible by 5.
15. How to check if a number is divisible by 3?
A number is divisible by 3 if the sum of its digits is divisible by 3. For instance, 21 is divisible by 3 because 2 + 1 = 3, which is divisible by 3.
