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)
The greatest common factor (GCF, also called greatest common divisor or GCD) is the largest number that divides evenly into two or more numbers, and it's the tool behind simplifying fractions to their lowest terms - reducing 24/36 to 2/3 requires finding that 12 is the GCF of both numbers. Beyond fractions, GCF is used for dividing groups of items into equal smaller groups without remainder (splitting 18 apples and 24 oranges into identical fruit baskets means finding the GCF of 18 and 24 to know the maximum number of baskets possible), and it underlies scheduling problems where events need to align at regular, evenly-divisible intervals.
The Euclidean algorithm, developed by the ancient Greek mathematician Euclid over 2,000 years ago, remains the standard efficient method for finding GCF, especially for larger numbers where simply listing all factors becomes impractical - it works by repeatedly applying division and remainder operations until reaching zero.
GCF finds the largest number that divides evenly into a set of numbers, while LCM (least common multiple) finds the smallest number that all the given numbers divide into evenly - they're mathematically related (GCF × LCM = the product of the two numbers, for any pair), but solve opposite kinds of problems.
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.
GCF finds the largest number that divides evenly into a set of numbers, while LCM finds the smallest number that all the numbers divide into evenly - they solve opposite kinds of problems.
Dividing both the numerator and denominator by their GCF reduces a fraction to its simplest form.
It's an efficient, ancient method for finding GCF by repeatedly applying division and remainder operations, still used today especially for larger numbers.
