Smallest $m$ for which $U_m$ has an element of order x?

Question

I am stuck on this question:

What is the Smallest $m$ for which $U_m$ has an element of order 13? Order 17? Order p? (p is prime)

The work I have tried so far:

I thought that maybe the element might be a generator, because they have the highest order in a group of units, but you can prove that in fact the element in $U_m$ with order p is not a generator, because a generator has order $\phi(m)$, and $\phi(m)$ can't be prime (because it is a product).

I don't really know what else to try.

Answer 1

Hint: If $p$ is a prime, then $U_m$ has an element of order $p$ iff $p$ divides $\phi(m)$, the order of $U_m$.

Now $\phi(m)= \prod p_i^{e_i-1}(p_i-1)$. Where can $13$ or $17$ or $p$ come from as a divisor of this product?