Given: Any non-negative pair of non-negative numbers can be represented as a single number which is the product of two specific exponentiations:
$ x = 2^a * 3^b $
To recover $a$, divide $x$ by 2, $i$ times, until it is no longer evenly divisible. To recover $b$, divide $x$ by 3, $j$ times, until it is no longer evenly divisible. [1]
I am having trouble deeply reasoning about why this works. Recovering a single exponent, e.g. $a$ from $2^a$ makes sense.
I am also having trouble asserting why the particular bases satisfy this. For generally recovery of $a$ and $b$, (constraining n to what can be represented by a and b), $a$ cannot be divisible by $b$, and vice versa. Intuitively this constraint makes sense, but I don't know if there are specific terms that address it.
Are there theorems or axioms for these properties? I am not necessarily looking to prove this myself, but to know what to reference to convince myself.
As a layperson, what terms should I be searching for to learn more? "Separable?"
[1] This is the answer to exercise 2.5 in the Structure and Interpretation of Computer Programs: https://web.mit.edu/6.001/6.037/sicp.pdf