It is usually desirable to compute the greatest common divisor of two numbers efficiently, one such method being the Euclidean Algorithm. What about inefficient methods to find the $\gcd$ of two numbers? Have I found one? Equivalent to the "fifth grade and up" method, we could construct the prime factorization of each number in various ways, e.g. iteratively dividing primes out, and then list the prime factors as two multisets (repeats are allowed). Then the $\gcd$ and $\text{lcm}$ would be the intersection and union respectively of the two multisets.
Example: $$ \text{f}(1890) =\{ 2,3,3,3,5,7\} \\ \, \\ \text{f}(510510) =\{2,3,5,7,11,13,17\} \\ \, \\ \gcd(1890,510510) = \text{f}(1890) \cap \text{f}(510510) =\{2,3,5,7\} \\ \, \\ \text{lcm}(1890,510510) = \text{f}(1890) \cup \text{f}(510510) =\{2,3,3,3,5,7,11,13,17\} $$ where $\text{f(n)}$ stands for the multiset of primes in the factorization of $\text{n}$.
Well, there's a reason one asks for the fastest way more than for the slowest way: you can always just stall for however long you like, and there's no well-defined notion of what counts as stalling and what's just a slow algorithm.
However, there is a very reasonable way to find the GCD that is decidedly slow. It's the greatest common divisor so... just list all of the divisors of each number, and find the biggest one they have in common. Pretty much no matter how you implement this, if you implement prime factorization with just as much care, listing all of the divisors is still a slower method.