Factor Calculator – Find All Factors, Prime Factors, and Factor Pairs

  • Adam
  • November 4, 2025

Free online factor calculator to find all factors, prime factorization, factor pairs, and check divisibility. Analyze any number with step-by-step solutions and comprehensive breakdown.

Factor Calculator

Advanced Tool for Finding Factors, Prime Factors, and Analyzing Divisibility

Quick Navigation

Find All Factors

List all positive divisors of a number

Prime Factorization

Break number into prime factors

Factor Pairs

Find pairs of factors that multiply to number

Check Divisibility

Test if number divides evenly

What are Factors?

A factor of a number is any positive integer that divides evenly into that number with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these divides 12 evenly. Every positive number has at least two factors: 1 and itself.

Factors are fundamental in mathematics and appear in many real-world situations. They're used when simplifying fractions, finding common multiples, organizing items into groups, and solving problems involving equal distribution. Understanding factors helps with divisibility tests, prime factorization, and computing greatest common factors and least common multiples.

There are several types of factors: all factors (complete list of divisors), prime factors (prime numbers that multiply to the number), and factor pairs (two numbers that multiply together to equal the original number). This calculator helps you find and analyze all types of factors with complete step-by-step explanations.

Key Concept: Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers. This is called prime factorization.

Key Features & Capabilities

This comprehensive factor calculator provides multiple analysis modes and detailed breakdown:

🔢 All Factors Find complete list of all positive divisors
🔍 Prime Factors Break numbers into prime factorization
🔗 Factor Pairs Find pairs that multiply to the number
✓ Divisibility Check Test if one number divides another
📊 Prime Checker Identify if number is prime or composite
📈 Factor Count Calculate total number of factors
📋 Step-by-Step Solutions Detailed breakdown showing all work
📊 Visual Organization Factors displayed in organized lists
📋 Copy to Clipboard One-click copy functionality
🎓 Educational Content Comprehensive guides and examples
⚡ Real-Time Calculation Instant results with no delays
📱 Fully Responsive Works on all devices seamlessly

How to Use This Calculator

Step-by-Step Guide

  1. Choose Analysis Type: Select the appropriate tab: All Factors (complete divisor list), Prime Factors (prime factorization), Factor Pairs (multiplication pairs), or Divisibility (division check).
  2. Enter Your Number: Input the positive integer you want to analyze. For divisibility checks, enter both the number and the divisor.
  3. Click Calculate: Press the Calculate button to perform the analysis using efficient algorithms.
  4. Review Results: The main result displays factors organized by type, with each factor clearly labeled.
  5. Study Steps: See detailed breakdown showing the method used and how factors were identified.
  6. Analyze Statistics: View count of factors, prime factorization, and related properties.
  7. Copy or Clear: Use Copy to transfer results. Use Clear to reset for a new calculation.

Tips for Accurate Use

  • Positive Integers Only: Enter only positive whole numbers. Zero and negative numbers don't have meaningful factors.
  • Large Numbers: The calculator works efficiently with large numbers using optimized algorithms.
  • Prime Numbers: Prime numbers have exactly two factors: 1 and themselves.
  • Factor Pairs: A number has fewer or equal factor pairs than total factors (depends on whether it's a perfect square).
  • Divisibility Test: A divisor either divides evenly (remainder 0) or doesn't divide at all.

Complete Formulas Guide

Finding All Factors

Test Each Number
For each i from 1 to √n:
If n mod i = 0, then i and n/i are both factors

Example: Find factors of 12
1 divides 12 → factors 1, 12
2 divides 12 → factors 2, 6
3 divides 12 → factors 3, 4
All factors: 1, 2, 3, 4, 6, 12

Prime Factorization

Repeated Division Method
Divide by smallest prime repeatedly until 1 remains

Example: Prime factorization of 60
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1

Result: 60 = 2² × 3 × 5

Factor Pairs

Multiplication Pairs
For each factor i, the pair (i, n/i) multiplies to n

Example: Factor pairs of 12
1 × 12 = 12
2 × 6 = 12
3 × 4 = 12

Pairs: (1,12), (2,6), (3,4)

Number of Factors

Count Using Prime Factorization
If n = p₁^a × p₂^b × p₃^c, then:
Number of factors = (a+1)(b+1)(c+1)

Example: 60 = 2² × 3¹ × 5¹
Factors = (2+1)(1+1)(1+1) = 3 × 2 × 2 = 12

Types of Factors Explained

All Factors (Divisors)

These are all positive integers that divide evenly into the number. For 24, all factors are 1, 2, 3, 4, 6, 8, 12, and 24. These are also called divisors. Every positive integer is always a factor of itself, and 1 is always a factor of every positive integer.

Prime Factors

Prime factors are the prime numbers that, when multiplied together, equal the original number. For 24 = 2³ × 3, the prime factors are 2 and 3. Prime factorization is unique for every number (Fundamental Theorem of Arithmetic). This representation is useful for finding GCF and LCM.

Factor Pairs

Factor pairs are two numbers that multiply together to equal the original number. For 24, the pairs are (1,24), (2,12), (3,8), and (4,6). If a number is a perfect square, one pair will be (√n, √n). The number of pairs equals half the total factors (rounded up).

Perfect Factors

These are factors that are perfect squares (1, 4, 9, 16, ...) or perfect cubes (1, 8, 27, ...). For 24, the perfect square factors are 1 and 4. These are useful in simplification and algebraic calculations.

Worked Examples

Example 1: Find All Factors of 24

Problem: List all factors of 24

Solution:
Test each number from 1 to √24 ≈ 4.9:
1 divides 24: 24 ÷ 1 = 24 → factors: 1, 24
2 divides 24: 24 ÷ 2 = 12 → factors: 2, 12
3 divides 24: 24 ÷ 3 = 8 → factors: 3, 8
4 divides 24: 24 ÷ 4 = 6 → factors: 4, 6

All factors: 1, 2, 3, 4, 6, 8, 12, 24
Total: 8 factors

Example 2: Prime Factorization of 60

Problem: Find prime factorization of 60

Solution:
Divide by smallest prime factors:
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1

Collecting: 60 = 2 × 2 × 3 × 5
60 = 2² × 3 × 5

Example 3: Factor Pairs of 36

Problem: Find all factor pairs of 36

Solution:
Find all factors first: 1, 2, 3, 4, 6, 9, 12, 18, 36

Create pairs:
1 × 36 = 36 → pair (1, 36)
2 × 18 = 36 → pair (2, 18)
3 × 12 = 36 → pair (3, 12)
4 × 9 = 36 → pair (4, 9)
6 × 6 = 36 → pair (6, 6)

Pairs: (1,36), (2,18), (3,12), (4,9), (6,6)

Example 4: Check if 7 is Prime

Problem: Determine if 7 is prime

Solution:
Find all factors of 7:
1 divides 7 ✓
2 divides 7? 7 ÷ 2 = 3.5 ✗
3 divides 7? 7 ÷ 3 ≈ 2.33 ✗
(Only need to check up to √7 ≈ 2.65)

Factors: 1, 7
7 is PRIME (exactly 2 factors)

Example 5: Count Factors Using Formula

Problem: Count factors of 72 = 2³ × 3²

Solution:
Prime factorization: 72 = 2³ × 3²

Using formula: (a+1)(b+1)(c+1)...
Number of factors = (3+1)(2+1) = 4 × 3 = 12

Verify: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Count: 12 ✓

Frequently Asked Questions

❓ Is 1 a prime number?
No. By definition, prime numbers have exactly two factors: 1 and themselves. Since 1 only has one factor (itself), it's neither prime nor composite. Primes start from 2.
❓ What's the difference between factors and multiples?
Factors divide into a number evenly (12 has factor 3 because 12 ÷ 3 = 4). Multiples are results of multiplying by a number (12 is a multiple of 3 because 3 × 4 = 12). They're inverse concepts.
❓ How many factors does 1 have?
1 has exactly one factor: itself. This is why 1 is neither prime nor composite. Every other positive integer has at least two factors.
❓ Can a number have an odd number of factors?
Yes, if it's a perfect square. Perfect squares like 9, 16, 25 have odd numbers of factors because one factor (the square root) pairs with itself. For 9: 1, 3, 9 (3 factors).
❓ What is prime factorization used for?
Prime factorization is used to: find GCF and LCM, simplify fractions, check divisibility, count factors, and solve problems in number theory. It reveals the fundamental structure of numbers.
❓ How do I find factors quickly for large numbers?
Only check up to √n. If i divides n, then both i and n/i are factors. This halves the checking required. For 100, only test 1-10, finding pairs like (1,100), (2,50), (4,25), (5,20), (10,10).
❓ When would I use factors in real life?
Factors are used in: organizing items into equal groups, creating rectangular arrays, finding common measurements, scheduling (when tasks repeat), simplifying fractions, and many construction/design problems.
❓ What is the unique factorization theorem?
The Fundamental Theorem of Arithmetic states every integer greater than 1 can be uniquely factorized into primes (ignoring order). For example, 12 = 2² × 3 is the only prime factorization.
❓ Can 0 have factors?
Technically yes, every non-zero number is a factor of 0. However, factors are typically defined only for positive integers, so 0 is usually excluded from factor discussions.

Start Finding Factors

Whether you're simplifying fractions, organizing items, analyzing numbers, or studying number theory, this comprehensive factor calculator provides instant solutions with complete analysis. Fast, accurate, and completely free.

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