What is the Greatest Common Factor?
The greatest common factor of two numbers is the largest number that evenly divides both numbers. For example, 221 and 425 have a greatest common factor of 17, because 13*17 = 221, and 17 * 25 = 425.
Euclidean Algorithm
This is an algorithm to get you the greatest common factor of two numbers.
This is the algorithm:
$$gcf(a,b) = gcf(b, a\bmod{}b)$$
$a \bmod b$ is just a better way of saying $a-b$. So, instead of saying $gcf(22, 3) = gcf(19,3) ... gcf(2, 3) = 0$, you could just say $gcf(22, 3) = gcf(3, 2) = 0$
Find the GCF of n! and n! + 3
$$gcf(n!, n! + 3) = gcf(n!, 3)$$ So, if $n \geq 3$, the gcf is 3. Otherwise, it's 1.
David Witten