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What Is the Greatest Common Factor?
The GCF (also called GCD — Greatest Common Divisor) is the largest number that divides evenly into two or more numbers. For example, GCF(24, 36) = 12, because 12 is the largest number that divides both 24 and 36 without a remainder.
Methods to Find the GCF
Method 1: List all factors of each number and find the largest common one. Method 2: Use prime factorization — take the lowest power of each shared prime factor. Method 3: Use the Euclidean algorithm — repeatedly divide and take remainders until the remainder is 0.
The Euclidean Algorithm
The Euclidean algorithm is the most efficient method: GCF(a,b) = GCF(b, a mod b). Repeat until the remainder is 0. Example: GCF(24,36) → GCF(36,24) → GCF(24,12) → GCF(12,0) → 12. This method has been used for over 2,300 years.
Applications of GCF
GCF is used to simplify fractions (divide numerator and denominator by their GCF), reduce ratios, distribute items equally, tile rectangular areas, and solve divisibility problems. If GCF(a,b) = 1, the numbers are coprime (no common factors).
Frequently Asked Questions
They are the same thing. GCF stands for Greatest Common Factor, and GCD stands for Greatest Common Divisor. GCF is more common in American math education, while GCD is used more in higher mathematics and internationally.
If GCF = 1, the numbers are called coprime or relatively prime. They share no common factors. For example, GCF(8, 15) = 1. Note that coprime numbers are not necessarily prime themselves.
Divide both numerator and denominator by their GCF. For example, 24/36: GCF(24,36) = 12, so 24/36 = (24÷12)/(36÷12) = 2/3.
No. The GCF is always less than or equal to the smallest number. It divides evenly into all the given numbers, so it cannot exceed any of them.