Greatest Common Factor (GCF) Calculator

Find the GCF of numbers or algebraic terms with a full, step-by-step solution.

Enter numbers (comma or space separated)

Enter algebraic terms below (e.g., 12x^2y, -18xy^3z)

The Ultimate Guide to the Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is the largest positive integer that divides a set of numbers without leaving a remainder. It's a fundamental concept in number theory and a critical first step in simplifying fractions and factoring polynomials. This guide will cover everything you need to know about GCF, from its basic definition to finding the GCF of complex algebraic terms.

What is the Greatest Common Factor?

For a set of integers, the GCF is the largest number that is a factor of every number in that set. For example, for the numbers 12 and 18, the factors of 12 are (1, 2, 3, 4, 6, 12) and the factors of 18 are (1, 2, 3, 6, 9, 18). The largest number that appears in both lists is 6, so the GCF is 6.

In algebra, the concept extends to monomials (terms with variables and exponents). The GCF of terms like 12x²y and 18xy³ involves finding the GCF of the coefficients (the numbers) and the GCF of the variables (the lowest power of each variable common to all terms).

How Does WebLab.Tools GCF Calculator Work?

Our calculator is a powerful tool designed to handle both numerical and algebraic GCF problems, providing a complete solution with steps.

GCF of Integers: For numbers, the calculator uses the prime factorization method. It breaks down each number into its prime factors and then identifies the common factors to multiply together.
GCF of Monomials: For algebraic terms, the calculator performs a two-part analysis. First, it finds the GCF of the numerical coefficients. Second, it analyzes the variables to find the highest power of each variable that can divide all terms. It then combines these to present the final GCF.

Benefits of Using WebLab.Tools GCF Calculator

Speed and Accuracy: Get instant, error-free results for complex sets of numbers or algebraic terms.
Handles Algebra: Our calculator is a GCF calculator with variables and exponents, making it perfect for algebra homework.
Step-by-Step Solutions: Don't just get the answer; learn the process. Our calculator provides a detailed breakdown of the prime factorization method, which is invaluable for students.
Handles Decimals: The calculator can even find the GCF of numbers with decimals by converting them to fractions and finding a common denominator.

How to Use WebLab.Tools GCF Calculator Effectively: A Pro's Guide

Example 1: Finding the GCF of Integers

Problem: Find the GCF of 48, 60, and 72.

Select the "GCF of Integers" tab.
Enter 48, 60, 72 into the text box.
Click "Calculate GCF".
Result: The GCF is 12. The solution will show the prime factorizations:
- 48 = 2 × 2 × 2 × 2 × 3
- 60 = 2 × 2 × 3 × 5
- 72 = 2 × 2 × 2 × 3 × 3
The common factors are 2 × 2 × 3, which equals 12.

Example 2: Finding the GCF of Monomials (Algebra)

Problem: Find the GCF of 12x²y and 18xy³.

Select the "GCF of Monomials" tab.
Enter 12x^2y into the first field and 18xy^3 into the second.
Click "Calculate GCF".
Result: The GCF is 6xy. The solution will show:
- The GCF of the coefficients (12 and 18) is 6.
- The lowest power of x common to both terms is x¹ (or x).
- The lowest power of y common to both terms is y¹ (or y).
- Combining these gives 6xy.

Frequently Asked Questions (FAQ)

How does this calculator find the GCF with steps?

Our calculator uses the prime factorization method. It breaks down each number in your set into its prime factors. The GCF is then found by multiplying all the prime factors that are common to every number in the set. The entire process is displayed in the solution section.

Can this GCF calculator handle variables and exponents?

Yes. Use the 'GCF of Monomials' tab. Enter your algebraic terms (e.g., 12x²y and 18xy³). The calculator finds the GCF of the coefficients (numbers) and the GCF of the variables (by taking the lowest power of each variable common to all terms) separately, then combines them for the final answer.

Does this work as a GCF calculator for polynomials?

This tool is specifically designed to find the GCF of monomials (single algebraic terms). Finding the GCF of a full polynomial involves factoring the entire expression, which is a more complex process. However, finding the GCF of the terms within the polynomial is often the first step in factoring, and our tool is perfect for that.

For finding the roots of a polynomial, try our Root Calculator. For simplifying fractions, which uses GCF, check out our Ratio Calculator.