Divisor Definition Formula Properties and Solved Examples
The concept of divisor in maths plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Knowing how to identify and work with divisors makes solving division, factors, and multiples much faster—an essential skill for competitive exams and daily problem-solving.
What Is Divisor in Maths?
A divisor in maths is a number that divides another number exactly, leaving zero as the remainder. For example, in 12 ÷ 3 = 4, the number 3 is the divisor. Divisors often appear in areas such as factors, multiples, and division problems. You’ll find this concept applied in arithmetic, number theory, and simplifying fractions.
Divisor, Dividend, Quotient and Remainder
| Term | Meaning | Example (42 ÷ 6 = 7) |
|---|---|---|
| Dividend | The number being divided | 42 |
| Divisor | Number that divides the dividend | 6 |
| Quotient | The result of the division | 7 |
| Remainder | What's left after division | 0 |
Key Formula for Divisor in Maths
Here’s the standard formula:
\( \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \)
To find the divisor, use:
\( \text{Divisor} = \frac{\text{Dividend} - \text{Remainder}}{\text{Quotient}} \)
Step-by-Step Illustration: How to Find Divisors of a Number
Let’s find all divisors of 18.
1. Start with the number 18.2. Begin checking from 1 upwards:
Does 1 divide 18? Yes (1 × 18 = 18)
Does 2 divide 18? Yes (2 × 9 = 18)
Does 3 divide 18? Yes (3 × 6 = 18)
Does 4 divide 18? No
Does 5 divide 18? No
Does 6 divide 18? Yes (6 × 3 = 18)
Does 9 divide 18? Yes (9 × 2 = 18)
Does 18 divide itself? Yes (18 × 1 = 18)
3. List of divisors: 1, 2, 3, 6, 9, 18.
Divisor Examples with Solutions
Example 1: Is 2 a divisor of 12?
2 divides 12 and leaves no remainder, because 12 ÷ 2 = 6. So, 2 is a divisor of 12.
Example 2: What are all divisors of 15?
1. List all numbers from 1 to 15 and check if they divide exactly.2. Dividing: 1 (yes), 3 (yes: 15 ÷ 3 = 5), 5 (yes: 15 ÷ 5 = 3), 15 (yes: 15 ÷ 15 = 1).
3. Answer: Divisors of 15 are 1, 3, 5, and 15.
Example 3: Find the divisor if Dividend = 56, Quotient = 8, Remainder = 0.
Use the formula: Divisor = (Dividend − Remainder) ÷ Quotient ⇒ (56 − 0) ÷ 8 = 7.
So, divisor is 7.
Divisor vs Dividend: Key Differences
| Divisor | Dividend |
|---|---|
| Divides another number | Is being divided |
| Usually smaller than or equal to dividend | Usually larger (except when equal or 1) |
| Appears after division sign (e.g., 12 ÷ 3) | Appears before division sign (e.g., 12 ÷ 3) |
Memory tip: “Dividend is divided, divisor does the dividing.”
Cross-Disciplinary Usage
Divisor in maths is not only useful in elementary arithmetic but also plays an important role in Physics (calculating rates), Computer Science (loops, mod), and daily logical reasoning. Students preparing for JEE or NTSE will often find questions on divisors, factors, and multiples in various patterns.
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: To check if a small number is a divisor of a large number, use divisibility rules. For example, to check if 6 divides 132:
- If the number is even and the sum of its digits is divisible by 3, then 6 is a divisor. 132 is even and 1+3+2=6 (which is divisible by 3), so 6 divides 132.
Apply similar tricks for quick calculations in timed exams. Vedantu’s live classes teach many such divisibility shortcuts and MCQ techniques.
Try These Yourself
- Write all divisors of 24.
- Is 7 a divisor of 56?
- List divisors of 36 between 1 and 10.
- Find a number which has exactly 2 divisors.
Frequent Errors and Misunderstandings
- Assuming divisor and factor always mean the same; remember, not every divisor is a factor when dealing with remainders.
- Forgetting to include the number itself and 1 as divisors.
- Mixing up dividend and divisor in equations.
Relation to Other Concepts
The idea of divisor in maths links directly with factors of a number and multiples. Mastering divisors also helps with advanced chapters like LCM and HCF, prime factors, and divisibility rules.
Classroom Tip
A quick way to remember “divisor divides, dividend is divided.” Draw a division bracket and always place the divisor outside, dividend inside. Vedantu’s expert teachers use such visual cues to make foundational arithmetic clear.
We explored divisor in maths—from definition, formula, examples, tricks, and differences. Practice more with Vedantu and related links to sharpen your concept and score better in school and entrance exams!
Related topics for practice: Divisibility Rules, Factors of a Number, LCM and HCF, Multiples, Prime Factors
FAQs on What Is a Divisor in Mathematics
1. What is a divisor in mathematics?
A divisor is a number that divides another number exactly without leaving a remainder. In other words, if a number a divides a number b such that b ÷ a gives a whole number, then a is a divisor of b.
- Example: 3 is a divisor of 12 because 12 ÷ 3 = 4 (no remainder).
- Here, 3 and 4 are both divisors of 12.
- Divisors are also called factors in arithmetic.
2. How do you find the divisors of a number?
To find the divisors of a number, divide the number by integers starting from 1 up to the number itself and check which divisions give no remainder.
- Step 1: Start with 1 and divide the number.
- Step 2: Continue dividing by 2, 3, 4, and so on.
- Step 3: List numbers that divide exactly.
3. What is the difference between a divisor and a multiple?
A divisor divides a number exactly, while a multiple is the result of multiplying a number by an integer.
- If 4 divides 20, then 4 is a divisor of 20.
- Since 4 × 5 = 20, 20 is a multiple of 4.
- Divisors go into the number; multiples come from the number.
4. What is a common divisor?
A common divisor is a number that divides two or more numbers exactly without leaving a remainder.
- Find divisors of each number separately.
- Identify the divisors that appear in both lists.
5. What is the greatest common divisor (GCD)?
The greatest common divisor (GCD) is the largest number that divides two or more numbers exactly.
- List all common divisors.
- Select the greatest one.
6. What is the formula to find the number of divisors of a number?
The number of divisors of a number can be found using its prime factorization and the formula (a + 1)(b + 1)(c + 1)... where exponents come from the prime powers.
- Step 1: Write the number as prime factors.
- Step 2: Add 1 to each exponent.
- Step 3: Multiply the results.
7. Is 1 a divisor of every number?
Yes, 1 is a divisor of every integer because every number divided by 1 equals itself.
- For any number n, n ÷ 1 = n.
- Therefore, 1 is always included in the list of divisors.
- This is true for all positive and negative integers.
8. What is a proper divisor?
A proper divisor is any divisor of a number except the number itself.
- List all divisors of the number.
- Remove the number from the list.
9. Can a divisor be negative?
Yes, a divisor can be negative because dividing by a negative number can still give an integer result.
- Example: 12 ÷ (−3) = −4.
- So −3 is also a divisor of 12.
- Both positive and negative divisors exist for integers.
10. What are the divisibility rules related to divisors?
Divisibility rules are shortcuts used to quickly check if a number is a divisor of another number without full division.
- Divisible by 2: Last digit is even.
- Divisible by 3: Sum of digits is divisible by 3.
- Divisible by 5: Last digit is 0 or 5.
- Divisible by 10: Last digit is 0.
