GCD & LCM Calculator

Calculate the Greatest Common Divisor (GCD) and Least Common Multiple (LCM) of up to 5 numbers, with prime factorization for each.

⚡ Quick Calculator Get a fast estimate

Number 1

Number 2

Number 3 (optional)

Number 4 (optional)

Number 5 (optional)

Results for: 12, 18

GCD = 6

LCM

GCD × LCM check

12 × 18 = 216

Prime Factorizations

122^2 × 3

182 × 3^2

How to Use

Enter 2 to 5 whole numbers. The calculator finds the GCD (also called HCF — Highest Common Factor) and LCM, showing the prime factorization of each number.

The Extended Calculator adds Euclidean algorithm steps, a Venn diagram of prime factors, and coprimality checking. The Professional Calculator includes the Extended Euclidean algorithm (Bezout's identity), multi-number GCD/LCM for up to 10 numbers, and Euler's totient function.

Need more detail?

📊 Extended Calculator More options, charts, and scenario comparison

First Number

Second Number

GCD(48, 18)

Factors of 48

2 × 2 × 2 × 2 × 3

Factors of 18

2 × 3 × 3

Prime Factor Venn Diagram

Euclidean Algorithm Steps

Step	Division	Remainder
1	48 = 2 × 18 + 12	12
2	18 = 1 × 12 + 6	6
3	12 = 2 × 6 + 0	0
GCD		6

GCD and LCM Formulas

GCD (Euclidean algorithm): gcd(a, b) = gcd(b, a mod b) until b = 0

LCM from GCD: lcm(a, b) = |a × b| / gcd(a, b)

GCD × LCM = a × b (for two numbers)

Bezout's Identity: There exist integers s, t such that gcd(a,b) = s·a + t·b

Worked Example

GCD and LCM of 12 and 18:

12 = 2² × 3 | 18 = 2 × 3²

GCD = 2 × 3 = 6

LCM = 2² × 3² = 36

Check: 6 × 36 = 216 = 12 × 18 ✓

Need full precision?

🔬 Professional Calculator Complete parameters, sensitivity analysis, and detailed breakdown

Find x, y such that: 252·x + 105·y = GCD(252,105)

Bezout's Identity

252·(-2) + 105·(5) = 21

GCD(252,105)

x (Bezout coefficient)

-2

y (Bezout coefficient)

Verification

21 = 21

Back-Substitution Steps

Step	a	b	Quotient	Remainder
1	252	105	2	42
2	105	42	2	21
3	42	21	2	0

Frequently Asked Questions

GCD is the largest number that divides all given numbers exactly. LCM is the smallest number that all given numbers divide into exactly. GCD simplifies fractions; LCM finds common denominators.

An efficient method to find the GCD. Repeatedly replace the larger number with the remainder: gcd(48, 18) → gcd(18, 12) → gcd(12, 6) → gcd(6, 0) = 6.

Adding or subtracting fractions with different denominators requires the LCM. To add 1/4 + 1/6, LCM(4, 6) = 12, so 3/12 + 2/12 = 5/12. LCM also finds when recurring events coincide.

Two numbers are co-prime if their GCD is 1 — they share no common factors other than 1. For example, 8 and 15 are co-prime. Co-prime numbers are fundamental in RSA encryption.

GCD & LCM Calculator

How to Use

GCD and LCM Formulas

Worked Example

Frequently Asked Questions

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