Originally Posted by

**tonio** What is $\displaystyle R^{*}\setminus U(R)$ , anyway? For me, $\displaystyle R^{*}$ usually denotes the units of R...but also $\displaystyle U(R)$!

Anyway: $\displaystyle d\mid a \Longrightarrow a=\prod\limits_{i=1}^tp_i^{a_i}=xd\,,\,\,x\in R\Longrightarrow\,\,\,if\,\,q\mid d\,,\,\,q\,\,a\,\,prime\,\,then\,\,q\mid p_i$ for some $\displaystyle 1\leq i\leq t$. Continue from here.

About $\displaystyle (a_1+1)\cdot...\cdot (a_t+1)$ : think of natural numbers: if a were a natural number, then the above product would be the number of positive divisors of a...take it from here.

Tonio