Quick answer: The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers, and the Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the GCF of 12 and 18 is 6, and the LCM of 12 and 18 is 36. Use the calculator below to find both instantly for any set of numbers.
Enter two or more numbers separated by commas to find their GCF and LCM.
The most reliable method for finding GCF by hand is the Euclidean algorithm, which repeatedly divides and takes remainders until you reach zero. The LCM can then be found using the relationship between GCF and LCM:
LCM(a, b) = (a x b) / GCF(a, b)
| Step | Calculation | Result |
|---|---|---|
| 1. Divide 18 by 12 | 18 = 1 x 12 + 6 | remainder 6 |
| 2. Divide 12 by 6 | 12 = 2 x 6 + 0 | remainder 0 |
| 3. GCF found | Last non-zero remainder | GCF = 6 |
| 4. Calculate LCM | (12 x 18) / 6 | LCM = 36 |
Another common approach is breaking each number into its prime factors, then comparing them.
| Number | Prime Factorization |
|---|---|
| 12 | 2 x 2 x 3 |
| 18 | 2 x 3 x 3 |
GCF uses the lowest power of each shared prime factor: common factors are 2 and 3, so GCF = 2 x 3 = 6.
LCM uses the highest power of every prime factor that appears in either number: 2 x 2 x 3 x 3 = 36.
| Numbers | GCF | LCM |
|---|---|---|
| 4, 6 | 2 | 12 |
| 8, 12 | 4 | 24 |
| 15, 20 | 5 | 60 |
| 9, 12, 15 | 3 | 180 |
| 10, 25 | 5 | 50 |
| 7, 9 | 1 | 63 |
GCF (Greatest Common Factor) is the largest number that divides evenly into a set of numbers, while LCM (Least Common Multiple) is the smallest number that all the numbers divide into evenly. They are opposite but related concepts.
The GCF of two different prime numbers is always 1, since prime numbers have no common factors besides 1.
The LCM of the denominators is used as the common denominator when adding or subtracting fractions with different denominators.
Yes. Calculate the GCF or LCM of the first two numbers, then repeat the process using that result and the next number, continuing until all numbers are included.
Yes, GCF (Greatest Common Factor) and HCF (Highest Common Factor) refer to the same concept and are used interchangeably depending on regional math curricula, with HCF more common in UK and Indian textbooks.
For any two numbers a and b, the product of their GCF and LCM equals the product of the two numbers: GCF(a,b) x LCM(a,b) = a x b.
If the only common factor is 1, the numbers are called coprime or relatively prime, and their LCM is simply the product of the numbers.
The Euclidean algorithm finds the GCF through repeated division in just a few steps, even for very large numbers, while listing all factors becomes slow and error-prone as numbers grow.
Students and quick manual calculations often run into a few recurring errors. The first is confusing GCF and LCM themselves, since both involve comparing shared factors between numbers; a helpful memory trick is that GCF is always less than or equal to the smallest number in the set, while LCM is always greater than or equal to the largest number.
The second common mistake is forgetting to use the lowest shared power of a prime factor for GCF and the highest power for LCM when using the prime factorization method. For example, with 8 (2 x 2 x 2) and 12 (2 x 2 x 3), the shared prime is 2, and the lowest shared power is 2 squared (4), not 2 cubed, so GCF = 4, not 8.
The third mistake is applying the two-number shortcut formula (LCM = a times b divided by GCF) directly to three or more numbers at once, which does not work reliably. Instead, calculate the LCM of the first two numbers, then find the LCM of that result and the third number, continuing one pair at a time.
GCF and LCM are foundational math concepts used everywhere from simplifying fractions to solving real-world scheduling and grouping problems. Use the calculator above to instantly find both for any set of numbers, and refer to the worked examples and reference table for quick manual checks.
