What is a Divisor in Maths?
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 Divisor in Maths: Definition, Formula, and Examples
1. What is a divisor in maths?
In maths, a divisor is a number that divides another number (the dividend) exactly, leaving no remainder. For example, 4 is a divisor of 12 because 12 ÷ 4 = 3 with no remainder. Understanding divisors is crucial for various mathematical concepts like factors, multiples, and prime factorization.
2. How do you find the divisors of a number?
To find all the divisors of a number, follow these steps:
- Start with 1 and the number itself. These are always divisors.
- Check for divisibility by 2, 3, 5, 7, and other prime numbers, up to the square root of the number.
- For each divisor you find, also include its corresponding factor (the result of the division).
- List all the divisors in ascending order.
3. What is the difference between a divisor and a dividend?
The dividend is the number being divided, while the divisor is the number doing the dividing. In the equation 12 ÷ 4 = 3, 12 is the dividend and 4 is the divisor. The result is the quotient (3 in this case). Sometimes, there's a remainder if the division isn't exact.
4. What are the divisors of 24?
The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. These are all the numbers that divide 24 without leaving a remainder.
5. Is 5 a divisor of 35?
Yes, 5 is a divisor of 35 because 35 ÷ 5 = 7 with no remainder.
6. Can a number be its own divisor?
Yes, every number is a divisor of itself. For example, 10 is divisible by 10 (10 ÷ 10 = 1).
7. What is the relationship between divisors and factors?
Divisors and factors are essentially the same thing. They are numbers that divide a given number without leaving a remainder. The terms are often used interchangeably.
8. How are divisors used in finding the greatest common factor (GCF)?
To find the greatest common factor (GCF) of two or more numbers, you first list the divisors of each number. Then, identify the largest divisor that is common to all the numbers. This largest common divisor is the GCF.
9. How do divisors relate to prime factorization?
Prime factorization involves expressing a number as a product of its prime divisors. For example, the prime factorization of 12 is 2 x 2 x 3. These prime numbers (2 and 3) are also divisors of 12.
10. What are common divisors, and how are they used?
Common divisors are numbers that are divisors of two or more numbers. Finding common divisors is important for simplifying fractions and calculating the greatest common factor (GCF).
11. Do negative numbers have divisors?
Yes, negative numbers also have divisors. For instance, the divisors of -12 include -1, -2, -3, -4, -6, and -12 (as well as 1, 2, 3, 4, 6, and 12).
12. How do I find divisors of a large number efficiently?
For large numbers, using a systematic approach is key. Start by checking for divisibility by small prime numbers (2, 3, 5, 7, etc.). You only need to check up to the square root of the number because any divisor larger than the square root will have a corresponding divisor smaller than it. Using prime factorization can also help significantly speed up the process.
