Show that the Kleene star of any unary language is regular
Let $p$ be the $\gcd$ of the lengths of all the words in $L$.
We claim that there exists $N$ such that for all $n&gt;N$, $1^n\in L$ iff $p|n$.

Observe that one direction is trivial: $p|n$ for every $1^n\in L$ by definition.

The converse is less trivial, you can find good guidelines in the answers here

Given this claim, you can write $L^*$ as the union of the relevant words up to length $N$, and then $1^{pi}$ for all large enough $i$.