About Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), of two or more numbers is the largest positive integer that divides each of the numbers without a remainder. It's a fundamental concept in mathematics with applications in fractions, algebra, and various real-world scenarios.

How to Calculate GCF

There are several methods to calculate the GCF:

Method 1: Using Prime Factorization

1. Find the prime factorization of each number.
2. Identify the common prime factors.
3. Multiply the common prime factors, using the lowest power they appear in any of the factorizations.

Method 2: Using the Euclidean Algorithm

For two numbers a and b (where a > b):

1. Divide a by b to get the quotient q and remainder r, so a = bq + r.
2. If r = 0, then GCF(a, b) = b.
3. Otherwise, GCF(a, b) = GCF(b, r).
4. Repeat until the remainder is 0.

Properties of GCF

• GCF(a, b) ≤ min(a, b)
• GCF(a, b) × LCM(a, b) = a × b
• If a divides b, then GCF(a, b) = a
• GCF(a, b, c) = GCF(GCF(a, b), c)

Applications of GCF

Field	Application
Mathematics	Simplifying fractions, solving Diophantine equations
Computer Science	Cryptography (RSA algorithm), error detection
Engineering	Gear design, signal processing
Everyday Life	Dividing items into equal groups, cutting materials efficiently

Example Problems

Example 1: Find the GCF of 36, 48, and 60.

Example 2: A carpenter has three boards of lengths 48 inches, 60 inches, and 84 inches. What is the longest possible length that can be used to measure each board an exact number of times?