Mathematical Tools
What It Does
Finds the greatest common divisor (GCD) and least common multiple (LCM) of a set of whole numbers, and gives each number’s prime factorization — showing the Euclidean-algorithm and trial-division steps. Everything runs in your browser.
How to Use It
- Enter your whole numbers, separated by commas, spaces, or newlines.
- Read the GCD, LCM, and each number’s prime factorization.
- Toggle Steps to see the working; Copy or export the results.
Options Explained
| Option | Description |
|---|---|
| Numbers | The whole numbers to analyze (up to 50). |
| Steps | Show the Euclidean-algorithm and trial-division working. |
| Factor limit | Largest prime factor shown individually; factors above this may be combined into a leftover. |
Enter one or more whole numbers to see the GCD, LCM, and prime factors.
About GCD, LCM & the Fundamental Theorem of Arithmetic
Every whole number bigger than 1 can be written as a product of prime numbers in exactly one way (ignoring order) — this is the fundamental theorem of arithmetic, and it’s the key to both the GCD and the LCM. The greatest common divisor of a set is the largest number that divides all of them; in prime terms it’s the product of the primes they share, each raised to the smallest power that appears. The least common multiple is the smallest number they all divide into; it’s the product of every prime that appears in any of them, each raised to the largest power.
The two are linked by a tidy identity — for a pair, GCD × LCM = a × b — which is why this tool can find the LCM straight from the GCD. The fastest way to get the GCD is the Euclidean algorithm, over two thousand years old: repeatedly replace the larger number by the remainder when it’s divided by the smaller, and the last non-zero value is the GCD (that’s why gcd(12,18) becomes gcd(18,12) → gcd(12,6) → gcd(6,0) = 6). Because this tool uses arbitrary-precision integers, it stays exact even for enormous numbers, and it runs entirely on your device.