Comprehensive Guide to GCF, LCM & Prime Factorization
The Greatest Common Factor (GCF), also known as the Greatest Common Divisor (GCD), is the highest whole number that evenly divides a set of numbers. The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of each number in the set.
Two-Number Relation: GCF(a, b) × LCM(a, b) = a × b
Euclidean Algorithm: gcd(a, b) = gcd(b, a mod b) until remainder is 0
Euclidean Algorithm: gcd(a, b) = gcd(b, a mod b) until remainder is 0
Methods for Computing GCF & LCM
- Prime Factorization Method: Break each number down into its prime base factors with exponents. For GCF, multiply the lowest power of every common prime. For LCM, multiply the highest power of all prime factors present.
- Euclidean Division Method: Highly efficient for large numbers. Successively divide and take the remainder until 0 is reached.
Frequently Asked Questions
What if two numbers have a GCF of 1?
Numbers whose greatest common factor is 1 are called coprime or relatively prime (for example, 8 and 15). In this case, their LCM is simply their product \(a \times b\).
Can GCF and LCM be calculated for more than two numbers?
Yes. You can compute \(\mathrm{GCF}(a, b, c) = \mathrm{GCF}(\mathrm{GCF}(a, b), c)\), and similarly for LCM.