Not so Flat After All

Any positive integer $v$ can be written as p_1^{a_1} p_2^{a_2} \ldots p_n^{a_n}, where p_i are prime numbers and a_i \ge 0. For example, 24 = 2^3 \times 3^1.

Pick any two distinct prime numbers p_1 and p_2. Imagine a two-dimensional plane where the powers of p_1 are plotted on the $x$-axis and the powers of p_2 are plotted on the $y$-axis. Any number that can be written as p_1^{a_1} p_2^{a_2} can be plotted on this plane at the location (x, y) = (a_1, a_2). The figure on the right shows a few examples where p_1 = 3 and p_2 = 2.

This idea can be extended to any $N$-dimensional space, where each of the $N$ axes is assigned a unique prime number. Each $N$-dimensional space has its own unique set of primes. We call such a set the Space Identification Set, or $S$ for short. $|S|$ (the cardinality of $S$) is $N$.

Any number that can be expressed as a product of the primes from $S$ (each raised to a non-negative integer power) can be plotted in this $|S|$-dimensional space. The figure at the bottom illustrates this idea for N = 3 and S = \{2, 3, 7\}. Clearly, any number plotted in space $A$ can also be plotted in space $B$ if S_A \subseteq S_B.

We define the distance between any two points in a given $N$-dimensional space as the sum of units travelled to go from one point to the other along the grid lines (i.e., movement is always parallel to one of the axes). For example, in the figure below, the distance between $168$ and $882$ is $4$.

Given two positive integers, write a program that determines the minimum order of a space where both numbers can be plotted, and the distance between these two integers in that space.