Factor Calculator - Find All Factors of Any Number Online
Find all factors, factor pairs, and identify perfect squares and cubes for any positive integer
Factor Calculator
All Factors:
Factor Pairs:
Summary
What is a Factor Calculator?
A Factor Calculator finds all factors of a number, displays factor pairs, and identifies special properties like perfect squares and cubes.
This calculator is perfect for:
- Students - Number theory and divisibility problems
- Teachers - Creating math exercises and examples
- Mathematicians - Analyzing number properties
- Quick Reference - Finding factors instantly
For prime factorization with exponential form and division steps, try our Prime Factorization Calculator to break down numbers into prime factors with detailed analysis.
For division problems with quotient and remainder calculations, use our Long Division Calculator to perform long division with step-by-step verification.
For basic arithmetic operations including addition and multiplication, check our Basic Calculator to perform simple calculations instantly.
To solve quadratic equations with detailed solutions, explore our Quadratic Equation Solver for finding roots using the quadratic formula.
How Factor Calculator Works
The calculator finds factors by testing divisibility:
Process:
- Test numbers from 1 to √n
- If f divides n evenly, both f and n/f are factors
- Collect all factors and sort them
- Create factor pairs
Example for 36:
Factors: 1, 2, 3, 4, 6, 9, 12, 18, 36
Key Factor Concepts
Factors
Integers that divide evenly into another number without leaving a remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4 exactly.
Factor Pairs
Two factors that multiply together to give the original number. For 12, the pairs are (1,12), (2,6), and (3,4).
Prime Factors
Factors that are prime numbers (divisible only by 1 and themselves). The prime factors of 12 are 2 and 3 (since 2 × 2 × 3 = 12).
Perfect Square
A number with an odd number of factors. This happens because one factor pair consists of the same number twice (e.g., 36 has 6×6).
Factors vs Multiples
It's easy to confuse factors and multiples, but they are opposites.
- Factors are numbers that fit inside the target number. They are finite. (Factors of 10: 1, 2, 5, 10)
- Multiples are numbers that the target number fits into. They are infinite. (Multiples of 10: 10, 20, 30, 40...)
Think of factors as the "ingredients" that make up the number, and multiples as the "products" the number can create.
Greatest Common Factor (GCF)
Finding factors is the first step to finding the Greatest Common Factor (GCF) of two or more numbers.
Why is GCF important?
- Simplifying Fractions: To reduce 24/36, you find the GCF of 24 and 36 (which is 12) and divide both top and bottom by it to get 2/3.
- Polynomial Factoring: In algebra, finding the GCF of terms (like 2x and 4x²) is the first step in factoring expressions.
Divisibility Rules
You can often spot factors without doing long division by using these tricks:
- • 2: The number is even (ends in 0, 2, 4, 6, 8).
- • 3: The sum of the digits is divisible by 3. (e.g., 123 -> 1+2+3=6, so 123 is divisible by 3).
- • 5: The number ends in 0 or 5.
- • 9: The sum of the digits is divisible by 9.
- • 10: The number ends in 0.
Real-World Applications
Arranging Objects: If you have 24 chairs, factors tell you all the possible rectangular arrangements: 1x24, 2x12, 3x8, 4x6. This is useful for event planning, planting gardens, or organizing marching bands.
Cryptography: Modern encryption (RSA) relies on the difficulty of factoring very large numbers. While multiplying two large primes is easy, finding those factors from the result is incredibly hard for computers, keeping your data safe.
How to Use This Calculator
Enter Number
Input any positive integer
Find Factors
Click to calculate all factors
View Results
See factors, pairs, and properties
Analyze
Check if prime, perfect square, or cube
Benefits of Using This Calculator
- • Complete Factor List: Shows all factors in ascending order.
- • Factor Pairs: Displays all factor pair combinations.
- • Special Properties: Identifies perfect squares and cubes.
- • Prime Detection: Instantly tells if number is prime.
Factors That Affect Your Results
1. Number Size
Larger numbers may have more factors but take slightly longer to calculate.
2. Prime Numbers
Prime numbers only have two factors: 1 and themselves.
3. Composite Numbers
Composite numbers have more than two factors.
4. Perfect Powers
Perfect squares and cubes have special factor patterns.
Frequently Asked Questions (FAQ)
Q: What are factors of a number?
A: Factors are whole numbers that divide evenly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Q: How do I find all factors of a number?
A: Test each number from 1 up to the square root of the target number. If it divides evenly, both it and its pair (number ÷ factor) are factors.
Q: What is a factor pair?
A: A factor pair is two numbers that multiply together to give the original number. For example, (3, 4) is a factor pair of 12 because 3 × 4 = 12.
Q: What is the difference between factors and prime factors?
A: Factors include all numbers that divide evenly into a number. Prime factors are only the prime numbers that divide into it. For example, 12 has factors 1,2,3,4,6,12 but prime factors 2 and 3.