Factorization Calculators

Factorization Calculator

Choose a calculator type and enter your numbers

Choose a calculator type, enter numbers, then press Calculate. Results appear instantly with step-by-step solutions.

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Results & Solutions

View your calculation results with step-by-step explanations

Result 1

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Result 2

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Step-by-Step Solution

Enter numbers and click Calculate to see the solution steps.

How It Works

Choose from three specialized calculators: LCM finds common multiples, GCF finds common factors, and Factor Calculator lists all divisors or prime factors of a number. Each provides step-by-step solutions.

Common Uses

Simplify fractions, solve ratio problems, schedule recurring events, factor polynomials, and find common denominators. Essential for algebra, number theory, and real-world scheduling.

Always Accessible

Works completely in your browser - no data sent to servers. Use it anytime, anywhere with full privacy protection for educational and professional use.

Factorization Formulas & Methods

LCM (Least Common Multiple) Calculation

LCM(a, b) = |a × b| ÷ GCD(a, b)

Where GCD is the Greatest Common Divisor. For multiple numbers: LCM(a, b, c) = LCM(LCM(a, b), c)

GCF (Greatest Common Factor) Calculation

Using Euclidean Algorithm: GCD(a, b) = GCD(b, a mod b)

For multiple numbers: GCD(a, b, c) = GCD(GCD(a, b), c). Also called Greatest Common Divisor (GCD).

Prime Factorization Method

n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ

Where p₁, p₂, ..., pₖ are prime numbers and a₁, a₂, ..., aₖ are their respective exponents (positive integers).

Step-by-Step Examples

Example 1: LCM of 12 and 18

Step 1: List multiples of 12: 12, 24, 36, 48... Step 2: List multiples of 18: 18, 36, 54, 72... Step 3: Find common multiples: 36, 72... Step 4: Smallest common multiple is 36

LCM(12, 18) = 36

Example 2: GCF of 24 and 36

Step 1: List factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Step 2: List factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Step 3: Find common factors: 1, 2, 3, 4, 6, 12 Step 4: Largest common factor is 12

GCF(24, 36) = 12

Example 3: Factors of 48

Step 1: Check divisibility from 1 to √48 Step 2: Find pairs: 1×48, 2×24, 3×16, 4×12, 6×8 Step 3: List all divisors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

48 has 10 factors

Understanding Factorization

Factorization is fundamental to mathematics, helping simplify complex problems, find common denominators, and solve real-world scheduling issues. Understanding LCM, GCF, and factors enables better problem-solving across algebra, number theory, and practical applications.

What is Factorization?

Factorization involves breaking down numbers into their component parts. LCM finds common multiples, GCF finds common divisors, and factor listing identifies all numbers that divide evenly into a given number.

Core Components of Factorization

Every factorization involves three key elements:

Prime Numbers: Numbers divisible only by 1 and themselves
Factors: Numbers that divide evenly into another number
Multiples: Numbers that result from multiplying a number by integers

How Prime Factorization Works

Every integer greater than 1 can be expressed uniquely as a product of prime numbers. This fundamental theorem of arithmetic underpins most factorization calculations and is essential for advanced mathematics.

Practical Example: Scheduling Events

Imagine you're scheduling three events: one every 4 days, one every 6 days, and one every 8 days. You need to find when all three events coincide.

LCM Calculation

Calculator Input:

Numbers: 4, 6, 8

Calculator Output:

LCM(4, 6, 8) = 24
Prime Factors: 4 = 2², 6 = 2 × 3, 8 = 2³
LCM = 2³ × 3 = 24

Interpretation

All three events coincide every 24 days. This shows how LCM solves real scheduling problems efficiently.

Frequently Asked Questions

What's the difference between LCM and GCF?

LCM (Least Common Multiple) finds the smallest number that is a multiple of all given numbers. GCF (Greatest Common Factor) finds the largest number that divides all given numbers without remainder.

Can I calculate LCM or GCF for more than two numbers?

Yes, our calculator supports multiple numbers. For LCM: LCM(a,b,c) = LCM(LCM(a,b),c). For GCF: GCF(a,b,c) = GCF(GCF(a,b),c). You can add as many numbers as needed.

What is prime factorization and why is it useful?

Prime factorization breaks a number down into its prime number components (e.g., 60 = 2² × 3 × 5). It's useful for finding LCM, GCF, simplifying radicals, and solving number theory problems.

How accurate is the calculator for large numbers?

The calculator handles numbers up to 1,000,000 accurately. For prime factorization of large numbers, it uses efficient algorithms but may take longer for numbers with large prime factors.

What if I enter negative numbers or decimals?

The calculator accepts only positive integers (1 to 1,000,000). Negative numbers, zero, decimals, and non-numeric inputs will trigger error messages with clear instructions.

Can I use this calculator for algebraic expressions?

This calculator works only with numerical values. For algebraic factorization (like factoring polynomials), you would need a different type of calculator or symbolic math software.

Factorization Calculators