What is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It is also known as the Greatest Common Divisor (GCD). For example, the GCF of 12 and 18 is 6, because 6 is the largest number that goes into both 12 and 18 evenly. Understanding the GCF is a foundational skill in mathematics, essential for simplifying fractions and solving various number theory problems. This calculator finds the GCF for you and demonstrates the step-by-step process using prime factorization.
How to Find the GCF
While you can find the GCF by listing all the factors of each number, the most efficient method for larger numbers is prime factorization:
- Find the Prime Factors: Break down each number into its prime factors. For example, to find the GCF of 48 and 60, the prime factors are `48 = 2⁴ × 3` and `60 = 2² × 3 × 5`.
- Identify Common Prime Factors: Look for all the prime factors that are present in the factorizations of *all* the numbers. In our example, both 48 and 60 share the prime factors 2 and 3.
- Find the Lowest Power: For each common prime factor, find the lowest power (exponent) it is raised to. In our case, the lowest power of 2 is `2²` (from 60) and the lowest power of 3 is `3¹` (from both).
- Multiply the Lowest Powers: Multiply these lowest powers together to get the GCF. For our example, `GCF = 2² × 3 = 4 × 3 = 12`.
Why is the GCF Important?
The most frequent and practical application of the GCF is **simplifying fractions**. To reduce a fraction to its simplest form, you divide both the numerator and the denominator by their GCF. For example, to simplify the fraction 12/18, you would find that the GCF of 12 and 18 is 6. Dividing both parts by 6 gives you 2/3. This concept is closely related to finding the Least Common Multiple (LCM), which is used to add and subtract fractions. You can explore this further with our LCM Calculator.
Beyond fractions, the GCF is useful for solving real-world problems that involve dividing things into smaller, equal groups. For example, if you want to create gift bags with an equal number of different items, the GCF will tell you the maximum number of identical bags you can create. For more practice and detailed explanations, the Khan Academy has excellent videos on the GCF.