How many isomorphism classes of associative rings (with identity) are there with 35 elements?
The underlying additive group $G$ is of the form $pq$ ($p=7, q=5$). So $G$ is either of form $\Bbb Z/35\Bbb Z$ or $C_7*C_5$. But what happens when they are a part of a ring. Will $C_7*C_5$ still a possibility with some appropriate multiplication.
There is only one ring up to isomorphism namely $\mathbb{Z}_{35}$. Since the ring is an abelian group of order $35$, the order of identity $1$ cannot be $5$ or $7$, as , if the order of $1$ is 5 then $1+1+1+1+1=0$ which implies for all $a\neq 0$ the order is $5$ as $a+a+a+a+a=a(1+1+1+1+1)=0$ which contradicts Cauchy's theorem of existence of an element of order $7$. Similarly the order of $1$ cannot be $7$. Hence the order of $1$ is $35$. Hence the elements of the ring are $1$, $1+1,\ldots,1+1+\cdots+1$. The distributive property ensures that $m.n=mn\pmod{35}$ where $m$ denotes addition of $m$ $1$'s. Hence the ring is isomorphic to $\mathbb{Z_{35}}$.
Edit: With the same reasoning we can prove that if $n$ is square free number then any ring( with identity) with $n$ elements is isomorphic to $\mathbb{Z_n}$.