What is the greatest common factor and least common multiple?

The greatest common factor (GCF), also called the maximum common divisor or highest common factor, is the largest number that divides evenly into all given numbers. The least common multiple (LCM) is the smallest number that is a multiple of all given numbers.

How to find the greatest common factor on a calculator using prime factorization?

To find the greatest common factor: 1) Prime factorize each number. 2) Identify the common prime factors. 3) Multiply the shared prime factors using the lowest exponents. For example, GCF(12, 18) = 2¹ × 3¹ = 6.

How to get the LCM of a number using the lcm calculator with steps?

To find the least common multiple: 1) Prime factorize each number. 2) Take each prime factor at its highest exponent. 3) Multiply together. For example, LCM(4, 6) = 2² × 3¹ = 12.

LCM Calculator with Steps & Prime Factorization

Greatest Common Factor & Least Common Multiple

Use our online least common multiple calculator and maximum common divisor calculator. Calculate the greatest common factor and least common multiple with full step-by-step prime factorization breakdown for any set of numbers.

First Number:

Second Number:

Quick Examples:

GCF / Maximum Common Divisor

Greatest Common Factor

LCM (Least Common Multiple)

144

Lowest Common Multiple

Verification (GCF × LCM = a × b): 12 × 144 = 48 × 36 = 1,728 ✓

LCM Calculator with Steps & Prime Factorization Breakdown

How to Find GCF & LCM — Step-by-Step Methods

Three methods for finding the greatest common factor and least common multiple: prime factorization, Euclidean algorithm (division method), and listing multiples for the lcm calculator with steps.

Prime Factorization Method

For GCF: multiply shared primes at lowest powers.

For LCM: multiply all primes at highest powers.

12 = 2² × 3

18 = 2 × 3²

GCF = 2¹×3¹ = 6

LCM = 2²×3² = 36

Euclidean Algorithm (Division)

Fastest method for large numbers. Repeatedly divide, using the remainder as the new divisor.

GCF(48, 36):

48 ÷ 36 = 1 R 12

36 ÷ 12 = 3 R 0

GCF = 12 ✓

LCM from GCF Formula

Once GCF is known, use the relationship: LCM(a,b) = (a × b) ÷ GCF(a,b).

LCM(48, 36):

= (48 × 36) / 12

= 1728 / 12

= 144 ✓

Least Common Multiple of Polynomials Calculator:

The least common multiple of polynomials calculator applies the same principle to algebraic expressions. Factor each polynomial completely, then construct the LCM using each unique factor at its highest degree. For example, LCM(x²−1, x²−2x+1) = LCM((x+1)(x−1), (x−1)²) = (x+1)(x−1)².

Our lcm calculator with variables and least common multiple calculator with variables handles integer coefficients — simply input the numeric coefficients of the leading terms to calculate their LCM or GCD.

Frequently Asked Questions

How to find the greatest common factor on a calculator with large numbers?

Our maximum common divisor calculator uses the Euclidean algorithm internally, which handles numbers up to 999,999 with nearly instant results — far faster than manually listing divisors for large numbers.

What is the difference between LCM and GCF?

The greatest common factor and least common multiple are complementary concepts. GCF answers "what is the biggest shared factor?" while LCM answers "what is the smallest shared multiple?" They're related by: GCF(a,b) × LCM(a,b) = a × b.

How to get the LCM of a number when there are more than two numbers?

To find the least common multiple — LCM of and across 3+ numbers, compute pairwise: LCM(a,b,c) = LCM(LCM(a,b), c). Our multi-number tab applies this chain automatically for up to 10 inputs, making it the ideal online least common multiple calculator for complex problems.