Find the Greatest Common Factor (GCF), also called Greatest Common Divisor (GCD), of two or more integers. Also shows LCM and Euclidean algorithm steps.
| Number | Prime Factors |
|---|---|
| 48 | 2^4 × 3 |
| 36 | 2^2 × 3^2 |
| 24 | 2^3 × 3 |
| GCF = 12 | 2^2 × 3 |
| Term | Meaning |
|---|---|
| GCF / GCD | Largest integer that divides all given numbers evenly |
| LCM | Smallest positive integer divisible by all given numbers |
| Co-prime | GCF = 1 (no common factors except 1) |
Step-by-step for the first two numbers.
| Step | Division | Remainder |
|---|---|---|
| 1 | 48 = 1 × 36 + 12 | 12 |
| 2 | 36 = 3 × 12 + 0 | 0 |
| Step | Calculation | Result |
|---|---|---|
| 1 | GCF(48, 36) | 12 |
| 2 | GCF(12, 24) | 12 |
| Property | Formula |
|---|---|
| For two numbers | GCF(a,b) × LCM(a,b) = a × b |
| LCM from GCF | LCM = (a × b) / GCF(a,b) |
| Euclidean step | GCF(a,b) = GCF(b, a mod b) |
Our free GCF calculator finds the Greatest Common Factor (also called the Greatest Common Divisor) and Least Common Multiple (LCM) of two or more numbers, showing step-by-step work using the Euclidean algorithm. It's a handy tool for students working on fraction simplification, factoring problems, and number theory homework.
The Euclidean Algorithm finds the GCF of two numbers by repeated division: divide the larger number by the smaller, then replace the larger number with the smaller number and the smaller number with the remainder, repeating until the remainder is 0. The last non-zero remainder is the GCF.
For more than two numbers, the GCF is found by calculating the GCF of the first two numbers, then finding the GCF of that result with the next number, and so on.
LCM (Least Common Multiple) is calculated using the relationship: LCM(a, b) = (a × b) ÷ GCF(a, b), since the product of two numbers always equals the product of their GCF and LCM.
The Greatest Common Factor (GCF) is the largest number that divides evenly into all given numbers. The Least Common Multiple (LCM) is the smallest number that all given numbers divide into evenly. They're related but solve opposite problems.
Dividing both the numerator and denominator of a fraction by their GCF reduces the fraction to its simplest form. For example, 24/36 simplifies to 2/3 by dividing both numbers by their GCF, 12.
The Euclidean algorithm is by far the fastest method for large numbers, since it uses repeated division rather than listing every factor, making it efficient even for numbers with many digits.
Yes, the GCF of multiple numbers can be found by calculating the GCF of two numbers at a time, working through the list sequentially, since GCF is associative, the order doesn't affect the final result.