I am having a little trouble understanding the following: 

"If $p\_1, \ldots, p\_k$ be the list of distinct primes dividing the product $mn,$ then we can factor $m$ and $n$ as $m=p\_1^{r\_1} \cdots p\_k^{r\_k}$ and $n=p\_1^{s\_1} \cdots p\_k^{s\_k},$ where $r\_i$ and $s\_i$ are nonnegative integers and may be zero in *at most one* of the decompositions for $m$ and $n.$" 

The italics are not mine, they are the authors. What is confusing me is indeed the `at most one' part. Is it saying:

1) that we may have $p^0$ only one time in either the $m$ decomposition OR the $n$ decomposition (but certainly not in both decompositions)? For example, $m=p\_1^0$ and $n=p\_1^1$.

2) that there can be a $p^0$ only one time in EACH of them (but basically appearing in both of them)? For example, $m=p\_1^0p\_2^1$ and $n=p\_1^1p\_2^0.$

3) that there can be a $p^0$ only one time per prime? I'm guessing that this is not the case that the authoer intended. That much seems obvious because if we had a distinct $p^0$ for both $m$ and $n$, then why bother mentioning the $p$ at all? For example, $m=2^0 \cdot p^1$ and $n = 2^0 \cdot p^2.$

If anyone wants any further context, just let me know. I'm working with the Fundamental Theorem of Finite Abelian Groups and factoring $\Bbb{Z}\_{mn}$ in various direct products $\Bbb{Z}\_{k\_1} \times \cdots \times \Bbb{Z}\_{k\_t}.$