How to determine if a number is divisible by 2,3,5,9, or 11
Divisibility rules are simple shortcuts that help us check whether a number can be divided evenly—without any remainder—by another number. These rules help make calculations faster and are useful in arithmetic, algebra, number theory, and even in daily life when making quick mental calculations.
In this article, you’ll learn the most important and commonly used divisibility rules for 2, 3, 5, 9, and 11, along with easy examples and practical applications.
What Is Divisibility?
A number A is divisible by another number B if:
- A ÷ B gives a whole number, and
- There is no remainder.
Example:
- 10 ÷ 2 = 5 → whole number → 10 is divisible by 2
- 10 ÷ 3 = 3 with remainder 1 → 10 is NOT divisible by 3
Understanding these rules helps simplify fractions, find factors, perform mental math, and solve more complex mathematical problems.
Divisibility Rules Explained
1. Divisibility by 2
A number is divisible by 2 if its last digit is even.
Ends with → 0, 2, 4, 6, 8
Examples:
- 14 → ends in 4 → divisible by 2
- 27 → ends in 7 → NOT divisible by 2
2. Divisibility by 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Examples:
- 123 → 1 + 2 + 3 = 6 → 6 is divisible by 3 → YES
- 124 → 1 + 2 + 4 = 7 → 7 is NOT divisible by 3 → NO
3. Divisibility by 5
A number is divisible by 5 if it ends in 0 or 5.
Examples:
- 25 → ends in 5 → divisible by 5
- 40 → ends in 0 → divisible by 5
- 26 → ends in 6 → NOT divisible by 5
4. Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
Example:
- 729 → 7 + 2 + 9 = 18 → 18 divisible by 9 → YES
If the digit sum is a multiple of 9 (9, 18, 27…), the number is divisible by 9.
5. Divisibility by 11
This rule uses alternating addition and subtraction.
Method:
- Add and subtract digits alternately.
- If the final result is 0 or a multiple of 11, the number is divisible by 11.
Example:
Number: 123456
Calculation:
1 − 2 + 3 − 4 + 5 − 6 = −3
Since −3 is not divisible by 11 → Not divisible by 11
Another example:
121 → 1 − 2 + 1 = 0 → divisible by 11
Practical Applications of Divisibility Rules
Divisibility rules are useful in several real-world situations:
1. Mental Math
Helps quickly check if numbers divide evenly without doing full division.
2. Simplifying Fractions
You can tell instantly whether the numerator and denominator have common factors.
3. Competitive Exams & School Tests
Almost every math exam includes questions where quick divisibility checks can save time.
4. Programming & Algorithms
Divisibility checks help optimise loops, conditions, and logical decisions.
5. Daily Finance
Useful while calculating bills, discounts, tax amounts, and splitting expenses.
Quick Practice Problems
Try applying what you learned:
- Is 54 divisible by 3?
✔ 5 + 4 = 9 → Yes - Is 121 divisible by 11?
✔ 1 − 2 + 1 = 0 → Yes - Is 275 divisible by 5?
✔ Ends in 5 → Yes - Is 842 divisible by 2?
✔ Ends in 2 → Yes - Is 738 divisible by 9?
✔ 7 + 3 + 8 = 18 → Yes
Conclusion
Divisibility rules make arithmetic faster, easier, and more intuitive. Whether you’re solving school math problems, preparing for a competitive exam, or doing everyday calculations, knowing these quick checks saves time and effort.
By understanding how to determine divisibility by 2, 3, 5, 9, and 11, you gain a strong foundation for handling more advanced mathematical concepts.