Factoring Calculator | Prime Factorization, GCF, LCM & Polynomials

Factoring Calculator

The all-in-one tool for factors, primes, GCF, LCM, and polynomials.

Enter a positive integer greater than 1.

Factor Order:
Details:
Try examples:

The Ultimate Factoring Calculator

Whether you are tackling algebra homework, preparing for an engineering exam, or solving complex number problems, this tool handles it all. Unlike basic calculators, our Factoring Calculator doesn't just give you the answer, it teaches you the math. Use it to instantly find factors, compute prime factorization with exponent notation, determine the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of multiple numbers, and even factor simple quadratic polynomials.

How to use the Factoring Calculator

1. Select your Mode

Choose what you want to calculate using the tabs at the top:

  • Factors List all integers that divide a number evenly.
  • Primes Break a number down into its basic prime building blocks.
  • GCF & LCM Compare multiple numbers to find common divisors and multiples.

2. Enter Numbers & Calculate

Type your number (e.g., 360). For GCF/LCM, separate numbers with commas (e.g., 12, 18, 30). For polynomials, use the standard math format like x^2 - 5x + 6.

3. Review Steps

Enable the "Show step-by-step process" checkbox to see the work. This is perfect for verifying your own math or learning the method behind the solution.

Example: Factoring the Number 360

Let's say you have a homework problem asking for the prime factorization of 360. Here is how you would use this tool to solve it:

  1. 1 Click the Prime Factorization tab at the top of the calculator.
  2. 2 Type 360 into the "Number to factor" field.
  3. 3 Ensure the "Show step-by-step process" box is checked.
  4. 4 Hit Calculate.
  5. The Result: You will see 2³ × 3² × 5 as the answer. The step-by-step area will show how 360 was divided by 2, then 2 again, until only prime numbers remained.

How This Calculator Works

This tool combines several mathematical algorithms to provide accurate results instantly. Here is the logic behind each mode:

Finding Factors

The calculator performs trial division. It iterates through integers from 1 up to the square root of your number. If a number divides evenly (leaving a remainder of 0), both the divisor and the resulting quotient are recorded as factor pairs.

Prime Factorization

We use an algorithm that repeatedly divides your number by the smallest possible prime number (starting with 2, then 3, 5, etc.) until the quotient is 1. The collection of these prime divisors forms the unique prime factorization string.

GCF & LCM Logic

For multiple numbers, the tool first computes the prime factorization for each input.
• GCF: Calculated by taking the lowest exponent of every common prime factor.
• LCM: Calculated by taking the highest exponent of every prime factor present across all numbers.

Polynomial Factoring

For quadratic expressions in the form ax² + bx + c, the calculator uses the AC Method. It searches for two integers that multiply to equal the product a×c and add up to equal the middle coefficient b. These numbers are then used to split the middle term and factor by grouping.

Factoring Examples: From Factors to Prime Factorization

Common numbers students encounter, broken down into their factors and prime building blocks.

Swipe table to see details
Number (n) All Factors of n Prime Factorization Count
12 1, 2, 3, 4, 6, 12 2² × 3 6
18 1, 2, 3, 6, 9, 18 2 × 3² 6
24 1, 2, 3, 4, 6, 8, 12, 24 2³ × 3 8
30 1, 2, 3, 5, 6, 10, 15, 30 2 × 3 × 5 8
36 1, 2, 3, 4, 6, 9, 12, 18, 36 2² × 3² 9
60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 2² × 3 × 5 12
100 1, 2, 4, 5, 10, 20, 25, 50, 100 2² × 5² 9

Understanding the Math: Factors, Primes, GCF & LCM

1. What are Factors & Prime Factorization?

Factors are the specific integers that divide a number evenly without leaving any remainder. You can think of them as the "building blocks" that multiply together to create that number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Prime Factorization digs deeper to find the "DNA" of a number. It breaks an integer down into a product consisting only of prime numbers (numbers divisible only by 1 and themselves, like 2, 3, 5, 7). According to the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization. This unique code is crucial for everything from simplifying fractions to modern computer cryptography.

Example (12):
Factors: 1, 2, 3, 4, 6, 12
Prime Factorization: 2 × 2 × 3 = 2² × 3

2. Understanding GCF and LCM

GCF (Greatest Common Factor), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more integers evenly. It is the most efficient tool for simplifying fractions to their lowest terms. For instance, if you have the fraction 4/6, finding the GCF (2) tells you to divide top and bottom by 2 to get the simplified fraction 2/3.

LCM (Least Common Multiple) is the smallest positive number that is a multiple of two or more integers. It represents the first time the "multiplication tables" of different numbers overlap. This is essential for finding the Least Common Denominator (LCD) when you need to add or subtract fractions with different bottoms.

Example (4 and 6):
Factors of 4: 1, 2, 4 | Multiples: 4, 8, 12, 16...
Factors of 6: 1, 2, 3, 6 | Multiples: 6, 12, 18, 24...
GCF: 2 (Largest shared factor)
LCM: 12 (First shared multiple)

Credible Sources & Further Reading

Frequently Asked Questions

What does it mean to factor a number?
Factoring a number means breaking it up into integers that can be multiplied together to get the original number. For example, 3 and 4 are divisors of 12 because 3 × 4 = 12. This mathematical process helps in simplifying expressions and solving equations, making complex problems easier to work with.

What is the difference between GCF and LCM?
The GCF (Greatest Common Factor) is the largest integer that divides all the numbers in your set evenly—it's used to simplify fractions to their lowest terms. The LCM (Least Common Multiple) is the smallest number that is a multiple of all the numbers in your set—it's used to determine common denominators when adding or subtracting fractions. For example, with 4 and 6: the GCF is 2 and the LCM is 12.

How do I use this online factoring calculator?
First, select your mode using the interactive interface tabs at the top (Factors, Prime Factorization, GCF & LCM, or Polynomials). Then input your number or numbers in the field. For GCF/LCM, separate multiple numbers with commas (e.g., 12, 18, 30). For polynomials, use standard math format like x^2 - 5x + 6 with variables and terms. Finally, click "Calculate" to generate instant results.

Can I see the step-by-step work for my solutions?
Yes! Simply enable the "Show step-by-step process" checkbox before clicking "Calculate." This educational feature is perfect for students who want to verify their homework, understand the methodology, or learn how to solve similar problems on their own with accurate guidance.

What is prime factorization and how does this tool display it?
Prime factorization breaks a number down into a product of only prime numbers (numbers divisible only by 1 and themselves). The calculator displays this using exponent notation for repeated primes. For instance, 12 is shown as 2² × 3, meaning 2 × 2 × 3. According to the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorization that this automated tool can identify.

Can this efficient calculator handle very large numbers?
Yes, this free online tool is optimized for performance and can compute standard integers safely up to 15 digits. Extremely large numbers (like those used in cryptography) might cause browser slowdowns, but for all standard high school and college algebra problems, the calculator provides instant and reliable results.

What kinds of polynomials can this calculator factor?
The calculator focuses on factoring quadratic trinomials in the form ax² + bx + c where the coefficients lead to integer or simple rational solutions. It uses the AC method to analyze and identify factor pairs that add up to the middle coefficient, making it an easy-to-use tool for common algebra problems.

How do I input multiple numbers correctly for GCF and LCM?
When using the GCF & LCM mode, enter your integers separated by commas (e.g., 12, 18, 24) or spaces in the input field. The dynamic calculator will automatically filter out duplicates and non-numeric characters before computing the result, so you don't need to worry about formatting errors.

Can I change the order in which factors are displayed in the interface?
Yes! The calculator includes a "Factor Order" option where you can choose between "Ascending" (smallest to largest) or "Descending" (largest to smallest) order for displaying your divisors. This visual flexibility helps you view the results in whichever format is most useful for your mathematical needs.

What numbers can I try as examples to test this tool?
The calculator suggests trying common examples like 36, 360, and 840 to see how it can solve various factorization problems. These numbers demonstrate different patterns—for instance, 360 has 24 factors and its prime factorization is 2³ × 3² × 5. You can also input any positive integer greater than 1 to calculate factors and determine prime factorization using this efficient, educational tool.

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