What is GCD (Greatest Common Divisor)?

The GCD, also known as HCF (Highest Common Factor), is the largest positive integer that divides all given numbers without a remainder. For example, GCD(12, 18) = 6 because 6 is the largest number that divides both 12 and 18 evenly.

What is LCM (Least Common Multiple)?

The LCM is the smallest positive integer that is divisible by all given numbers. For example, LCM(4, 6) = 12 because 12 is the smallest number that both 4 and 6 divide into evenly.

What is the relationship between LCM and GCD?

For two numbers a and b: LCM(a, b) × GCD(a, b) = a × b. This relationship allows you to find LCM if you know GCD, and vice versa. For example, LCM(12, 18) × GCD(12, 18) = 36 × 6 = 216 = 12 × 18.

What is the Euclidean Algorithm?

The Euclidean Algorithm is an efficient method to find the GCD. It works by repeatedly dividing the larger number by the smaller and taking the remainder, until the remainder is 0. The last non-zero remainder is the GCD.

How do you find LCM using prime factorization?

To find LCM using prime factorization: 1) Find prime factors of each number, 2) Take each prime factor with its highest power among all numbers, 3) Multiply them together. For GCD, take each common prime factor with its lowest power.

What are common uses of LCM and GCD?

GCD is used for simplifying fractions, finding common denominators, and solving problems involving equal distribution. LCM is used for finding when events coincide (like scheduling), adding fractions with different denominators, and solving rate problems.

LCM & GCD Calculator

Calculate the Least Common Multiple (LCM) and Greatest Common Divisor (GCD/HCF) of two or more numbers. See step-by-step solutions using Euclidean algorithm and prime factorization.

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calculate LCM & GCD Calculator

Enter Numbers

Enter 2 or more positive integers

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Parsed Numbers

12, 18

Key Relationship

LCM × GCD = a × b

For two numbers a and b

functions Results

GCD (Greatest Common Divisor)

Also called HCF (Highest Common Factor)

LCM (Least Common Multiple)

Smallest number divisible by all inputs

Verification

GCD × LCM = 6 × 36 = 216 = 12 × 18 ✓

Prime Factorization

12 = 2² × 3

18 = 2 × 3²

How it works:

GCD: Take lowest powers → 2¹ × 3¹ = 6

LCM: Take highest powers → 2² × 3² = 36

Euclidean Algorithm (GCD)

18 = 12 × 1 + 6

12 = 6 × 2 + 0

→ GCD = 6

lightbulb Tips

•LCM × GCD = a × b
•GCD = HCF (same thing)
•Use Euclidean algorithm
•GCD=1 means coprime

calculate Reference

Common GCDs

GCD(12,18)=6GCD(24,36)=12 GCD(15,25)=5GCD(48,60)=12

Key Formulas

LCM × GCD = a × b

LCM = (a × b) / GCD

How to Use This Calculator

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Enter Numbers

Input two or more positive integers separated by commas.

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Choose Calculation

Select LCM, GCD, or both calculations.

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Select Method

View Euclidean algorithm, prime factorization, or both.

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View Results

See LCM/GCD with step-by-step solution.

The Formula

GCD is the largest number that divides all given numbers. LCM is the smallest number divisible by all given numbers.

LCM(a,b) × GCD(a,b) = a × b

lightbulb Variables Explained

GCD Greatest Common Divisor
LCM Least Common Multiple
HCF Highest Common Factor (same as GCD)

tips_and_updates Pro Tips

GCD (Greatest Common Divisor) is also called HCF (Highest Common Factor)

LCM × GCD = Product of the two numbers (for 2 numbers)

If GCD = 1, the numbers are coprime (no common factors)

Use GCD to simplify fractions: divide numerator and denominator by their GCD

LCM is useful for finding common denominators when adding fractions

The Euclidean algorithm efficiently finds GCD by repeated division