How to Find Factors of a Number in Maths
The concept of factors of a number is central to mathematics. It helps students split numbers into equal groups, solve division-based problems, and is foundational for topics like HCF, LCM, and prime factorization. Understanding factors makes calculations faster and builds strong number sense for exams and practical life.
What Is a Factor of a Number?
A factor of a number is a whole number that divides the given number exactly, with no remainder. For example, 4 is a factor of 12 because 12 ÷ 4 = 3 (no remainder). Factors are also called divisors in maths. They are the building blocks of a number and help us in prime factorization, HCF, LCM, and simplifying fractions.
Key Formula for Factors of a Number
To find the total number of factors of a number, use this formula:
If N = \( p_1^{a_1} \times p_2^{a_2} \times \ldots \times p_k^{a_k} \) (prime factorization), then
Total number of factors = (a₁+1) × (a₂+1) × ... × (ak+1)
Cross-Disciplinary Usage
The idea of finding factors of a number is useful in maths, physics (for even measurements and groupings), computer science (loops, algorithms), and day-to-day logic. Factors help in coding exercises where divisibility is checked, in competitive exams, and in real-world tasks like dividing things evenly among groups.
Step-by-Step Illustration: How to Find Factors of a Number
- Start with 1 and the number itself. (1 and the number are always factors.)
- Test each number from 2 up to the square root of the number:
- List all unique factors - write in ascending order for clarity.
Quick Example: Factors of 24
Try dividing 24 by 1, 2, 3, 4, 6:
24 ÷ 1 = 24
24 ÷ 2 = 12
24 ÷ 3 = 8
24 ÷ 4 = 6
So, factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
Examples: Common Factors, Prime Factors, and Factor Pairs
| Number | Factors | Prime Factors | Factor Pairs |
|---|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 2, 3 | (1,12), (2,6), (3,4) |
| 18 | 1, 2, 3, 6, 9, 18 | 2, 3 | (1,18), (2,9), (3,6) |
| 24 | 1, 2, 3, 4, 6, 8, 12, 24 | 2, 3 | (1,24), (2,12), (3,8), (4,6) |
Check detailed solutions for specific numbers in our guides: Factors of 12, Factors of 18, Factors of 24.
Speed Trick or Vedic Shortcut
To find factors of large numbers quickly:
- Do prime factorization first (break the number into products of primes).
- Use the total factors formula for instant counting.
Example: Find total factors of 60 (60 = 2² × 3 × 5).
Total factors = (2+1) × (1+1) × (1+1) = 3×2×2 = 12 factors.
Tip: Write factors in pairs to avoid missing any!
Difference between Factors and Multiples
| Factors | Multiples |
|---|---|
| Divide the number exactly | Number is obtained by multiplying |
| Always smaller or equal | Always greater or equal |
| Finite for each number | Infinite for each number |
| Eg: Factors of 6: 1,2,3,6 | Eg: Multiples of 6: 6,12,18,24,... |
Try These Yourself
- List all the factors of 28.
- How many factors does 45 have?
- Is 9 a factor of 72?
- Identify all the prime factors of 100.
Common Mistakes and Misunderstandings
- Confusing factors with multiples (“Is 20 a factor of 5?” – No, 5 is a factor of 20.)
- Forgetting to include 1 or the number itself as a factor.
- Missing out factor pairs, especially for bigger numbers.
- Counting a factor twice (e.g. perfect squares like 25: 5 × 5 is just one factor, 5).
Relationship to Other Maths Concepts
Knowing the factors of a number helps with higher topics like HCF and LCM, fraction simplification, algebraic identities, and divisibility rules. If you understand factors, you are well prepared for chapters on prime numbers, prime factorization, and factors and multiples.
Classroom Tip
To quickly list factors, check divisibility up to the square root and write the divisor and quotient together (they come in pairs). For MCQs, use prime power count formula for instant answers — a tip Vedantu teachers use in live online classes!
We have explored factors of a number—definition, methods, formulas, examples, quick tips, and their connections with other topics. Regular practice at Vedantu will strengthen your number skills and boost your exam preparation. Ready for more? Try related concepts like Prime Factorization or Multiples to master the number world!
FAQs on Factors of a Number Explained with Examples
1. What are factors of a number in Maths?
In mathematics, the factors of a number are whole numbers that divide the number exactly, leaving no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 without leaving a remainder. Understanding factors is crucial for mastering concepts like highest common factor (HCF) and lowest common multiple (LCM).
2. How do you find the factors of a number?
There are several ways to find the factors of a number. One method is to systematically test each whole number from 1 up to the number itself, checking if it divides the number evenly. Another efficient technique involves finding prime factorization, expressing the number as a product of prime numbers. Once you have the prime factorization, you can list all possible combinations of these prime factors to find all factors.
3. What is the difference between factors and multiples?
Factors are numbers that divide a given number without leaving a remainder. Multiples are the results of multiplying a given number by any whole number. For example, the factors of 6 are 1, 2, 3, and 6, while the multiples of 6 are 6, 12, 18, 24, and so on.
4. How do you find the number of factors of a number?
To find the total number of factors, first find the prime factorization of the number (e.g., 24 = 2³ × 3¹). Then, add 1 to each exponent in the prime factorization (3 + 1 = 4 and 1 + 1 = 2). Finally, multiply these results together (4 × 2 = 8). Therefore, 24 has 8 factors.
5. What are the factors of 24?
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
6. What are the factors of 12?
The factors of 12 are 1, 2, 3, 4, 6, and 12.
7. How to find factors of a big number?
Finding factors of large numbers is best done using prime factorization. Start by dividing the number by the smallest prime numbers (2, 3, 5, 7, etc.) until you reach 1. This gives you the prime factors. Then, use the method described in question 4 to find the total number of factors or systematically list all combinations of these prime factors.
8. What is the prime factorization of a number?
Prime factorization is the process of expressing a number as a product of its prime factors. A prime number is a whole number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7). For example, the prime factorization of 12 is 2 × 2 × 3 or 2² × 3.
9. How are factors used to find the HCF (Highest Common Factor)?
To find the HCF of two or more numbers, find the prime factorization of each number. The HCF is the product of the common prime factors raised to the lowest power present in any of the factorizations. For example, to find the HCF of 12 (2² × 3) and 18 (2 × 3²), the common prime factors are 2 and 3. The lowest power of 2 is 2¹, and the lowest power of 3 is 3¹. Therefore, the HCF is 2 × 3 = 6.
10. How are factors used in solving word problems?
Factors are used in many word problems involving division, sharing, or grouping. For example, if you have 24 apples and want to divide them equally among a certain number of people, the possible numbers of people are the factors of 24 (1, 2, 3, 4, 6, 8, 12, 24), because only these numbers divide 24 without leaving any remainder.
11. Can you explain the concept of factor pairs?
Factor pairs are two numbers that multiply together to give a specific number. For example, the factor pairs of 12 are (1, 12), (2, 6), and (3, 4). Understanding factor pairs can make it easier to find all factors of a number efficiently.
12. How is the square root used when finding factors?
When finding all factors of a number, you only need to test whole numbers up to the square root of that number. If a number divides the original number evenly, its corresponding factor will be larger than the square root. For example, to find factors of 100, test numbers only up to 10 (√100 = 10).
