Find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of up to 10 numbers with step-by-step Euclidean algorithm.
Euclidean Algorithm for GCF:
For two numbers a and b (a > b):
GCF(a, b) = GCF(b, a mod b)
Repeat until b = 0, then GCF = a
For multiple numbers:
GCF(a, b, c) = GCF(GCF(a, b), c)
LCM using GCF:
LCM(a, b) = (a × b) / GCF(a, b)
LCM(a, b, c) = LCM(LCM(a, b), c)
GCF (Greatest Common Factor), also called GCD (Greatest Common Divisor), is the largest positive integer that divides all given numbers without a remainder. It's useful for simplifying fractions and solving problems involving divisibility.
LCM (Least Common Multiple) is the smallest positive integer that is a multiple of all given numbers. It's used when finding common denominators for fractions, scheduling recurring events, or solving problems involving periodic phenomena.
The Euclidean algorithm repeatedly replaces the larger number with the remainder of dividing it by the smaller number. This process continues until the remainder is zero. The last non-zero remainder is the GCF. It's one of the oldest known algorithms.
The GCF of zero and any number n is n (the non-zero number). The LCM of zero and any number is undefined (or zero, depending on definition). Our calculator handles zero appropriately in GCF calculations.
Yes, the calculator works with negative numbers. The GCF is always positive (by convention). The LCM is also positive. The algorithm uses absolute values for calculations, so -12 and 18 have the same GCF as 12 and 18.
