Every journey through tree space is a sequence of prime operations.
From any number to any other, there's a shortest path through multiplication and division.
The shortest path from a to b always passes through their GCD. First, we divide out factors unique to a. Then we multiply in factors unique to b.
path(a, b) = a → gcd(a,b) → b
The path length equals the tree distance: the total number of prime factor differences. This reveals that GCD isn't just a number — it's the intersection point of two numbers in tree space.