jmc

For $n$ to have an inverse $(\mathrm{mod}130)$, it is necessary for $n$ to be relatively prime to 130. Conversely, if $n$ is relatively prime to 130, then $n$ has an inverse $(\mathrm{mod}130)$. The same holds for 231. Therefore we are looking for the smallest positive $n$ that is relatively prime to 130 and 231.

We can factor $130=2\cdot 5\cdot 13$ and $231=3\cdot 7\cdot 11$. These are all of the primes up to 13, so none of the integers $2-16$ is relatively prime to both 130 and 231. However, 17 is relatively prime to both of these numbers. So the smallest positive integer greater than 1 that has a multiplicative inverse modulo 130 and 231 is $\boxed{17}$.