Two methods, and when each is easier
Prime factorisation is what schools teach because it shows why the answer is what it is: break every number into primes, then keep only what they all have in common. It is clear, and it becomes painful once the numbers get large — factoring a six-digit number by hand is real work.
The Euclidean algorithm does not care how big the numbers are. Divide the larger by the smaller, keep the remainder, repeat with the pair you just used. The last non-zero remainder is the GCF. It reaches the answer for two enormous numbers in a handful of lines, which is why it is what computers actually use.
GCF, GCD, HCF — same thing
Greatest common factor, greatest common divisor and highest common factor are three names for one idea, differing mainly by country and textbook. They always give the same number.
The everyday use is reducing fractions: divide the top and bottom by their GCF and you land in lowest terms in one move rather than several.
Frequently asked questions
What if the numbers have nothing in common?
Then the GCF is 1 and the numbers are called coprime. It does not mean either of them is prime — 8 and 9 are coprime, and neither is.
Can I find the GCF of more than two numbers?
Yes, and this page does. The GCF of a whole list is found by taking the GCF of the first two, then that answer with the third, and so on.
Is anything uploaded?
No. The numbers stay in the page and nothing is sent anywhere.
Something wrong with this tool, or an idea for it? Tell us