Factoring Calculator

Find all factors and factor pairs of any number with step-by-step solutions. Perfect for homework, learning divisibility, and understanding number theory.

Factoring Calculator

Find the Factors of:

Solution:

Enter a number and click Calculate to see all factors and factor pairs

What are Factors?

Factors (also called divisors) are whole numbers that divide evenly into another number with no remainder. In other words, when you divide a number by one of its factors, the result is always a whole number. Every positive integer has at least two factors: 1 and itself.

Understanding factors is fundamental to number theory and is essential for learning about divisibility, prime numbers, greatest common factors (GCF), least common multiples (LCM), simplifying fractions, and solving many types of algebraic equations.

Example:

The factors of 12 are: 1, 2, 3, 4, 6, and 12

This is because 12 can be divided evenly by each of these numbers:

12 ÷ 1 = 12
12 ÷ 2 = 6
12 ÷ 3 = 4
12 ÷ 4 = 3
12 ÷ 6 = 2
12 ÷ 12 = 1

How to Find Factors

There are several methods to find all factors of a number:

Method 1: Division Method

Test each number from 1 up to the number you're factoring:

1. Start with 1 (always a factor)
2. Try dividing by 2, 3, 4, 5, etc.
3. If the division results in a whole number, both the divisor and quotient are factors
4. Continue until you reach the number itself

Example: Finding factors of 20

20 ÷ 1 = 20 → factors: 1, 20

20 ÷ 2 = 10 → factors: 2, 10

20 ÷ 4 = 5 → factors: 4, 5

All factors: 1, 2, 4, 5, 10, 20

Method 2: Pair Method (More Efficient)

Only test numbers up to the square root of the number:

1. Find the square root of the number
2. Test only numbers from 1 to the square root
3. When you find a factor, also record its pair (number ÷ factor)
4. This reduces the number of tests needed

Example: Finding factors of 36

√36 = 6, so test 1 to 6

36 ÷ 1 = 36 → 1 and 36

36 ÷ 2 = 18 → 2 and 18

36 ÷ 3 = 12 → 3 and 12

36 ÷ 4 = 9 → 4 and 9

36 ÷ 6 = 6 → 6 (pair with itself)

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

Method 3: Prime Factorization

Break the number into its prime factors, then find all combinations:

1. Find the prime factorization of the number
2. List all possible combinations of the prime factors
3. Each combination is a factor

Example: Finding factors of 24

24 = 2³ × 3¹

Possible combinations:

2⁰ × 3⁰ = 1

2¹ × 3⁰ = 2

2² × 3⁰ = 4

2³ × 3⁰ = 8

2⁰ × 3¹ = 3

2¹ × 3¹ = 6

2² × 3¹ = 12

2³ × 3¹ = 24

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

Factor Pairs

Factor pairs are two numbers that multiply together to give the original number. Every factor has a corresponding pair factor. Finding factor pairs is useful for understanding multiplication, division, and for working with rectangular arrays in geometry.

How to Find Factor Pairs:

1. Start with 1 and the number itself (always a pair)
2. Find the next smallest factor
3. Divide the number by that factor to find its pair
4. Continue until factors start repeating

Example: Factor pairs of 48

1 × 48 = 48

2 × 24 = 48

3 × 16 = 48

4 × 12 = 48

6 × 8 = 48

Types of Numbers Based on Factors

Prime Numbers

Numbers with exactly 2 factors: 1 and itself.

Examples:

2 (factors: 1, 2)

3 (factors: 1, 3)

5 (factors: 1, 5)

7 (factors: 1, 7)

11 (factors: 1, 11)

Composite Numbers

Numbers with more than 2 factors.

Examples:

4 (factors: 1, 2, 4)

6 (factors: 1, 2, 3, 6)

8 (factors: 1, 2, 4, 8)

9 (factors: 1, 3, 9)

10 (factors: 1, 2, 5, 10)

Perfect Squares

Numbers with an odd number of factors.

Examples:

1 (factors: 1)

4 (factors: 1, 2, 4)

9 (factors: 1, 3, 9)

16 (factors: 1, 2, 4, 8, 16)

25 (factors: 1, 5, 25)

Applications of Factors

Mathematics

• Simplifying fractions
• Finding GCF and LCM
• Solving algebraic equations
• Prime factorization
• Number theory problems
• Divisibility rules
• Modular arithmetic

Real-World Uses

• Arranging objects in equal groups
• Dividing items evenly
• Creating rectangular arrays
• Scheduling and time management
• Resource allocation
• Music theory (rhythm divisions)
• Computer science algorithms

Problem Solving

• Area and perimeter problems
• Rectangular garden layouts
• Tile arrangement patterns
• Packaging and containers
• Team formation
• Time interval calculations
• Measurement conversions

Advanced Topics

• Cryptography and security
• Number patterns and sequences
• Optimization problems
• Graph theory
• Combinatorics
• Digital signal processing
• Error detection codes

Common Factor Patterns

Number	Factors	Count	Type
12	1, 2, 3, 4, 6, 12	6	Composite
24	1, 2, 3, 4, 6, 8, 12, 24	8	Composite
36	1, 2, 3, 4, 6, 9, 12, 18, 36	9	Perfect Square
48	1, 2, 3, 4, 6, 8, 12, 16, 24, 48	10	Composite
60	1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60	12	Highly Composite
100	1, 2, 4, 5, 10, 20, 25, 50, 100	9	Perfect Square

Frequently Asked Questions (FAQ)

What is the difference between factors and multiples?

Factors divide into a number evenly (12 ÷ 3 = 4, so 3 is a factor of 12), while multiples are the result of multiplying a number by whole numbers (12 × 3 = 36, so 36 is a multiple of 12). Factors are smaller than or equal to the number, while multiples are larger than or equal to the number.

Can negative numbers have factors?

Yes, negative numbers have factors. The factors of -12 include both positive and negative divisors: ±1, ±2, ±3, ±4, ±6, ±12. However, when we talk about factors in elementary mathematics, we typically refer only to positive factors for simplicity. This calculator shows positive factors.

What is the fastest way to find all factors?

The most efficient method is to test only numbers up to the square root of the number. For example, for 100, you only need to test 1 through 10. When you find a factor (like 2), you automatically know its pair (100 ÷ 2 = 50). This cuts the work roughly in half and is the method used by this calculator.

How many factors does a prime number have?

A prime number has exactly 2 factors: 1 and itself. This is the defining characteristic of prime numbers. For example, 7 has only two factors (1 and 7), 13 has only two factors (1 and 13), and 29 has only two factors (1 and 29). The number 1 is special—it has only one factor (itself) and is neither prime nor composite.

Why do perfect squares have an odd number of factors?

Perfect squares have an odd number of factors because one factor pairs with itself. For example, 36 = 6 × 6. All other factors come in pairs (1 and 36, 2 and 18, 3 and 12, 4 and 9), but 6 pairs with itself, giving 36 a total of 9 factors (an odd number). This pattern applies to all perfect squares.

What number has the most factors?

There's no single answer—larger numbers tend to have more factors, but it depends on the number's prime factorization. Numbers with many small prime factors have more divisors. For example, 120 has 16 factors, and 840 has 32 factors. Highly composite numbers are defined as having more factors than any smaller positive integer.

How do I find the GCF using factors?

To find the Greatest Common Factor (GCF) of two numbers, list all factors of each number, then identify the largest factor that appears in both lists. For example, factors of 12 are 12 and factors of 18 are 18. The common factors are 6, so the GCF is 6.

Can 0 or 1 have factors?

Zero is divisible by every number (except zero itself), so it has infinitely many factors. The number 1 has exactly one factor: itself. Because 1 only has one factor rather than two or more, it is classified as neither prime nor composite—it's a special case called a unit in number theory.

Need Help with Math Concepts?

Our expert tutors can help you master factors, divisibility, prime numbers, and all math topics

Book Your Free Demo Session