Factors of a Number Calculator

Factors of a Number Calculator — Find All Factors & Factor Pairs

Find all factors and factor pairs of any integer with step-by-step explanation. Learn about proper factors, prime factorization, perfect numbers, and how to quickly list all divisors of a number. For related number theory tools, see our prime factorization calculator.

Table of Contents

1. 1What Are Factors?
2. 2How to Find All Factors
3. 3Step-by-Step Examples
4. 4More Factor Examples
5. 5Factor Pairs
6. 6Proper Factors and Perfect Numbers
7. 7Prime Factors vs. All Factors
8. 8How to Count Factors
9. 9Common Mistakes to Avoid
10. 10Real-World Applications
11. 11Practice Problems with Solutions
12. 12Tips for Finding Factors
13. 13Final Thoughts
14. 14FAQ: Common Questions

Find All Factors

Enter a positive integer to find all its factors and factor pairs. Click Calculate for a complete list with step-by-step working.

Enter a Number

Factors will appear here.

Note: A factor (or divisor) of a number is an integer that divides evenly into that number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Every number has at least two factors: 1 and itself.

---

Factors of a Number – Complete Explanation

A factor (also called a divisor) of a number is an integer that divides evenly into that number — meaning there is no remainder. Understanding factors is fundamental to working with fractions, finding common denominators, simplifying expressions, and exploring the structure of numbers in number theory.

Factor Definition

A factor of n is an integer that divides n evenly (with remainder 0).

If a × b = n, then both a and b are factors of n.

Every positive integer greater than 1 has at least two factors: 1 and itself.

1. What Are Factors?

Factors come in pairs — when you find one factor, you automatically find its partner. For example, if you discover that 3 is a factor of 36 (because 36 ÷ 3 = 12 with no remainder), then 12 must also be a factor of 36. This pairing property makes finding all factors much more efficient.

Key terminology:

• Factor: Any integer that divides the number evenly.
• Factor Pair: Two factors whose product equals the number.
• Proper Factor: A factor other than the number itself.
• Prime Factor: A factor that is also a prime number.
• Common Factor: A factor shared by two or more numbers.

2. How to Find All Factors

The systematic method for finding all factors of a number n:

1. Start with 1 — it's always a factor.
2. Test each integer from 2 up to √n:
   • If n ÷ i has no remainder, then i and n/i are both factors.
   • You only need to test up to √n because any factor larger than √n will be paired with one smaller than √n.
3. List all factors in order from smallest to largest.
4. n itself is always the largest factor.

Why stop at √n? If n = a × b, then at least one of a or b must be ≤ √n. If both were > √n, their product would exceed n. So by checking up to √n, you guarantee finding every factor pair.

3. Step-by-Step Examples

Example 1: Finding Factors of 36

Find all factors of 36.

√36 = 6, so we only need to check divisors 1 through 6.

1:  36 ÷ 1 = 36  → factors: 1 and 36
2:  36 ÷ 2 = 18  → factors: 2 and 18
3:  36 ÷ 3 = 12  → factors: 3 and 12
4:  36 ÷ 4 = 9   → factors: 4 and 9
5:  36 ÷ 5 = 7.2 (not a factor)
6:  36 ÷ 6 = 6   → factor: 6 (pairs with itself)

All factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Total: 9 factors (odd because 36 is a perfect square)

Example 2: Finding Factors of 48

Find all factors of 48.

√48 ≈ 6.93, so check divisors 1 through 6.

1:  48 ÷ 1 = 48  → 1 and 48
2:  48 ÷ 2 = 24  → 2 and 24
3:  48 ÷ 3 = 16  → 3 and 16
4:  48 ÷ 4 = 12  → 4 and 12
5:  48 ÷ 5 = 9.6 (not a factor)
6:  48 ÷ 6 = 8   → 6 and 8

All factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Total: 10 factors (even because 48 is not a perfect square)

4. More Factor Examples

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 → 8 factors

Factors of 17: 1, 17 → 2 factors (prime number)

Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 → 9 factors (perfect square)

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 → 12 factors

5. Factor Pairs

Every factor belongs to a factor pair — two numbers that multiply to give the original number. Listing factor pairs is an organized way to ensure you haven't missed any factors.

Factor pairs of 36:

1 × 36 | 2 × 18 | 3 × 12 | 4 × 9 | 6 × 6

Since 6 appears twice (6 × 6), 36 has an odd number of total factors (9).

Factor pairs of 48:

1 × 48 | 2 × 24 | 3 × 16 | 4 × 12 | 6 × 8

All pairs are distinct — 48 has an even number of total factors (10).

6. Proper Factors and Perfect Numbers

Proper factors are all factors of a number except the number itself. They're used to classify numbers:

Perfect Number: Sum of proper factors = the number itself.

Example: 6 → proper factors: 1, 2, 3 → 1+2+3 = 6 (perfect!)

Example: 28 → proper factors: 1, 2, 4, 7, 14 → sum = 28 (perfect!)

Abundant Number: Sum of proper factors > the number.

Example: 12 → proper factors: 1, 2, 3, 4, 6 → sum = 16 > 12 (abundant)

Deficient Number: Sum of proper factors < the number.

Example: 10 → proper factors: 1, 2, 5 → sum = 8 < 10 (deficient)

7. Prime Factors vs. All Factors

Prime factors are the building blocks — they're the prime numbers that multiply to give the original number. All factors include every possible divisor, both prime and composite.

For 36:

Prime factors: 2, 2, 3, 3 → 2² × 3²

All factors: 1, 2, 3, 4, 6, 9, 12, 18, 36

Every factor is a product of the prime factors with exponents ranging from 0 to their maximum.

8. How to Count Factors Without Listing Them

If you know the prime factorization, you can calculate the number of factors:

Factor Count Formula

If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ

Number of factors = (a₁ + 1)(a₂ + 1)...(aₖ + 1)

Example: 36 = 2² × 3²

Number of factors = (2 + 1)(2 + 1) = 3 × 3 = 9 factors

Example: 48 = 2⁴ × 3¹

Number of factors = (4 + 1)(1 + 1) = 5 × 2 = 10 factors

9. Common Mistakes to Avoid

• Forgetting 1 and the number itself: 1 and n are always factors. Don't leave them out.
• Missing the second factor in a pair: When you find one factor, always write down its partner (the quotient).
• Checking all the way to n instead of √n: You only need to check up to the square root — everything beyond that comes from pairs.
• Listing duplicates: For perfect squares, the square root only counts as one factor, not two.
• Confusing factors with multiples: Factors of 12 are 1, 2, 3, 4, 6, 12. Multiples of 12 are 12, 24, 36, 48...

10. Real-World Applications

• Simplifying Fractions: Finding common factors to reduce fractions to simplest form.
• Arranging Objects: Finding factors tells you all the ways to arrange items in rectangular arrays (e.g., 36 chairs in 4 rows of 9).
• Division Problems: Knowing factors helps with mental division and checking divisibility.
• Cryptography: The difficulty of factoring large numbers is the basis of RSA encryption.
• Music: Rhythmic patterns often involve factors and multiples of time signatures.
• Construction: Cutting materials into equal lengths requires finding factors of the total length.

11. Practice Problems with Solutions

Problem 1: Find all factors of 30.

Solution: 1, 2, 3, 5, 6, 10, 15, 30 → 8 factors

Problem 2: Find all factors of 49.

Solution: 1, 7, 49 → 3 factors (perfect square: 7²)

Problem 3: How many factors does 72 have?

Solution: 72 = 2³ × 3² → (3+1)(2+1) = 4×3 = 12 factors

Problem 4: Is 28 a perfect number?

Solution: Proper factors: 1, 2, 4, 7, 14 → sum = 28 → Yes, perfect!

12. Tips for Finding Factors Quickly

• Always start with 1 and the number itself — they're guaranteed.
• Use divisibility rules to quickly test small numbers.
• Work systematically from 2 up to √n — don't skip numbers.
• For each factor found, immediately write down its paired factor.
• Perfect squares have an odd number of factors; all other numbers have an even number.
• If a number is prime, its only factors are 1 and itself.

13. Final Thoughts

Finding the factors of a number is a foundational mathematical skill that connects to fractions, divisibility, prime numbers, and algebraic factoring. The systematic approach — testing divisors up to the square root and recording both members of each factor pair — ensures you find every factor efficiently and accurately.

Understanding factors deeply prepares you for working with greatest common factors, least common multiples, simplifying rational expressions, and exploring the rich structure of the integers. Use this calculator to verify your manual work, but practice finding factors until the process becomes systematic and reliable.

14. Frequently Asked Questions

A factor divides evenly into a number (e.g., factors of 12: 1, 2, 3, 4, 6, 12). A multiple is the result of multiplying a number by an integer (e.g., multiples of 12: 12, 24, 36, 48...). Factors are ≤ the number; multiples are ≥ the number.

Factors come in pairs. If n = a × b, at least one of a or b must be ≤ √n. If both were larger than √n, their product would exceed n. By checking divisors up to √n, you find one member of every factor pair. The partner is automatically found by division.

A proper factor is any factor of a number except the number itself. For example, the proper factors of 12 are 1, 2, 3, 4, and 6 (excluding 12). Proper factors are used to classify numbers as perfect, abundant, or deficient.

A perfect number is a number where the sum of its proper factors equals the number itself. The first two perfect numbers are 6 (1+2+3=6) and 28 (1+2+4+7+14=28). Perfect numbers are rare — the next one is 496.

Use prime factorization. If n = p₁ᵃ¹ × p₂ᵃ² × ... × pₖᵃᵏ, then the number of factors is (a₁+1)(a₂+1)...(aₖ+1). For example, 36 = 2² × 3², so it has (2+1)(2+1) = 9 factors.

Factors come in pairs that multiply to the number. For perfect squares, the square root pairs with itself (e.g., 6 × 6 = 36). This self-pairing means one factor is counted once instead of twice, resulting in an odd total. All other numbers have factors in distinct pairs, giving an even total.

RSA encryption relies on the mathematical difficulty of factoring very large numbers. While multiplying two large primes is computationally easy, finding the original primes from their product is extremely difficult. This asymmetry is what keeps encrypted communications secure.

Explore Math Topics

Continue your learning journey with our other math guides

🔢 Basic Arithmetic 📐 Pre-Algebra 📊 Algebra 📏 Geometry 📈 Statistics

Related Calculators

Explore these other math tools that complement factor calculations:

• 🔹 Prime Factorization Calculator — Break down numbers into prime factors.
• 🔹 GCD Calculator — Find the greatest common divisor of numbers.
• 🔹 LCM Calculator — Find the least common multiple of numbers.
• 🔹 Prime Number Checker — Check if a number is prime or composite.
• 🔹 Divisibility Checker — Test divisibility rules for any number.
• 🔹 Perfect Number Checker — Check if a number is perfect, abundant, or deficient.

Conclusion

Finding the factors of a number is a foundational mathematical skill that connects to fractions, divisibility, prime numbers, and algebraic factoring. Use this calculator to verify your manual work, but practice finding factors until the process becomes systematic and reliable.

Readers who wish to continue practicing and reinforce what they've learned can explore the companion learning resource for guided exercises.