Localizations of $\mathbb{Z}/m\mathbb{Z}$ Let $A=\mathbb{Z}/m\mathbb{Z}$. Prove that for each multiplicative subset $\Sigma$ of $A$ there is an integer $n$ such that $\Sigma^{-1}A=\mathbb{Z}/n\mathbb{Z}$.
 A: Ok. First of all some facts about fractions: 
Let $R$ be a commutative with identity. A subset $S$ of $R$ is called saturated if for any $a,b \in R$, $ab \in S \longleftrightarrow a,b \in S$. Now if $S$ is a multiplicative closed subset of $R$ the intersection of all saturated subsets of $R$ containing $S$ is denoted by $\bar{S}$ (called the saturation of $S$). The importance of this notion is the following two theorems:
Theorem 1. A subset $S$ of $R$ is saturated if anf only if $R\backslash S$ is a union of prime ideals. 
Theorem 2. For any multiplicative closed subset $S$ of $R$ we have $S^{-1}R \cong {\bar{S}}^{-1}R$. 
This means that when working with isomorphisms you can always assume that $S$ is saturated (by Theorem 2) and saturated subsets are more well formed (by Theorem 1). On the other hand the following is also useful: 
Theorem 3. Suppose $R_1, R_2$ are rings and let $S$ be a saturated subset of $R_1\times R_2$. Then there are saturated subsets $S_1$ in $R_1$ and $S_2$ in $R_2$ such that $S = S_1\times S_2$. (The converse is also clear)
Theorem 4.  Suppose $R_1, R_2$ are rings and let $S_1$ be a multiplicative subset of $R_1$ and $S_2$ be a multiplicative subset of  $R_2$ then $(S_1\times S_2)^{-1}(R_1\times R_2) \cong {S_1}^{-1}R_1\times {S_2}^{-1}R_2$. 
Now lets return to your problem. Suppose $ p_1^{\alpha_1}\times\cdots\times p_k^{\alpha_k}$ is a factorization of $m$ into distinct primes. Then $A=\mathbb{Z}/m\mathbb{Z} \cong \Bbb{Z}_m \cong \Bbb{Z}_{p_1^{\alpha_1}}\times\cdots\times\Bbb{Z}_{p_k^{\alpha_k}}$. Now aplly Theorems $3,4,1,2$. 
Feel free to ask for more explanation. 
A: Hint: what can you say about the image of a generator of $A$ under the natural map $A \to \Sigma^{-1}A$?
Edit: Hint 2: Can you do the case when $n = p^k$ is a prime power? For the general case, consider the Chinese Remainder Theorem.
