# GCD and LCM calculator

> Greatest common divisor (GCD, GCF or HCF) and least common multiple (LCM) of two or more whole numbers, with Euclid's algorithm step by step.

Interactive version: https://www.calcopenly.com/math/gcd-lcm-calculator
Subject: Math calculators

The greatest common divisor (GCD, also called the greatest common factor or highest common factor) is the largest whole number that divides every input; the least common multiple (LCM) is the smallest positive number that every input divides. Euclid's algorithm finds the GCD by repeatedly replacing the larger number with its remainder after division by the smaller: gcd(1071, 462) = gcd(462, 147) = gcd(147, 21) = 21. The LCM follows from lcm(a, b) = |a × b| ÷ gcd(a, b), applied a pair at a time.

Reducing fractions uses the GCD; adding fractions and lining up repeating schedules use the LCM. For the default 12, 18 and 30 the GCD is 6 and the LCM is 180, so events every 12, 18 and 30 days next coincide after 180 days.

For two numbers the calculator also gives the Bézout identity from the extended Euclidean algorithm: 21 = 1071 × (−3) + 462 × 7. Negative inputs count by their absolute value, gcd(0, n) = n, and a list containing 0 has an LCM of 0.

## Inputs

- **Whole numbers**: Two or more integers, separated by commas or spaces.

## Results

- Greatest common divisor — main result
- Least common multiple
- Bézout identity

## Formula

$$
\begin{gathered} \gcd(a, b) = \gcd(b,\ a \bmod b) \\[6pt] \gcd(a, 0) = |a| \\[10pt] \operatorname{lcm}(a, b) = \frac{|a\,b|}{\gcd(a, b)} \end{gathered}
$$

## Worked examples

### 12, 18 and 30

- Whole numbers: 12, 18, 30
- **Greatest common divisor: 6**
- **Least common multiple: 180**
- Checked against: Python 3.8 math.gcd and a·b // gcd folded over the list

### Euclid's 1071 and 462

- Whole numbers: 1071, 462
- **Greatest common divisor: 21**
- **Least common multiple: 23,562**
- **Bézout identity: 21 = 1071·(−3) + 462·7**
- Checked against: Wikipedia — Euclidean algorithm worked example (gcd 21); lcm and Bézout coefficients by hand back-substitution, checked in Python

### Extended Euclid on 240 and 46

- Whole numbers: 240 46
- **Greatest common divisor: 2**
- **Least common multiple: 5,520**
- **Bézout identity: 2 = 240·(−9) + 46·47**
- Checked against: Wikipedia — Extended Euclidean algorithm example table (s = −9, t = 47)

### Coprime 17 and 31

- Whole numbers: 17, 31
- **Greatest common divisor: 1**
- **Least common multiple: 527**
- Checked against: Both prime, so gcd 1 and lcm 17 × 31 = 527

### Zero with 5 (edge)

- Whole numbers: 0, 5
- **Greatest common divisor: 5**
- **Least common multiple: 0**
- **Bézout identity: 5 = 0·0 + 5·1**
- Checked against: gcd(0, n) = n and lcm(0, n) = 0 by definition (Python math.gcd(0, 5) = 5)

### Negative input −12 and 18

- Whole numbers: -12, 18
- **Greatest common divisor: 6**
- **Least common multiple: 36**
- Checked against: Python math.gcd(-12, 18) = 6; lcm = |−12·18| / 6

## Questions

### How do you find the GCD of two numbers?

Use Euclid's algorithm: divide the larger number by the smaller, keep the remainder, and repeat with the divisor and the remainder until the remainder is 0; the last non-zero remainder is the GCD. For 1071 and 462: 1071 = 2 × 462 + 147, 462 = 3 × 147 + 21, 147 = 7 × 21 + 0, so the GCD is 21. By Lamé's theorem it never needs more than five steps per digit of the smaller number.

### How do you find the LCM of two numbers?

Divide their product by their GCD: lcm(a, b) = a × b ÷ gcd(a, b). For 12 and 18, gcd = 6, so lcm = 216 ÷ 6 = 36. For more numbers, go one pair at a time: lcm(36, 30) = 1,080 ÷ 6 = 180, so the LCM of 12, 18 and 30 is 180. Listing multiples (12, 24, 36 …) reaches the same answer, but slowly for large numbers.

### What is the difference between GCD, GCF and HCF?

There is none: greatest common divisor (GCD), greatest common factor (GCF) and highest common factor (HCF) are three names for the same number. GCF and HCF are the usual school terms, GCF mostly in the US and HCF in the UK and India, while GCD is the name in number theory and programming, as in Python's math.gcd. Under every name, gcd(12, 18) = 6.

### How are the GCD and LCM related?

For two positive whole numbers, gcd(a, b) × lcm(a, b) = a × b; with 12 and 18, 6 × 36 = 216 = 12 × 18. In prime factors, the GCD takes the lower power of each shared prime and the LCM the higher power of every prime: 12 = 2² × 3 and 18 = 2 × 3², so the GCD is 2 × 3 = 6 and the LCM is 2² × 3² = 36. The product rule does not extend to three or more numbers.

### What does it mean when two numbers are coprime?

Their GCD is 1, so they share no prime factor, and their LCM is simply their product. 17 and 31 are coprime, with an LCM of 17 × 31 = 527, and so are 8 and 15 although neither is prime. A fraction is in lowest terms exactly when its numerator and denominator are coprime.

### How accurate is the GCD and LCM calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “12, 18 and 30” is checked against Python 3.8 math.gcd and a·b // gcd folded over the list.

### Where does the method come from?

Euclid, Elements, Book VII, Propositions 1–2; Knuth, The Art of Computer Programming Vol. 2, §4.5.2 (Euclid's algorithm); Wikipedia — Extended Euclidean algorithm (worked example 240, 46).

## Sources

- Euclid, Elements, Book VII, Propositions 1–2
- Knuth, The Art of Computer Programming Vol. 2, §4.5.2 (Euclid's algorithm)
- [Wikipedia — Extended Euclidean algorithm (worked example 240, 46)](https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm)