Prove or Disprove the following statemnet

Question

Prove or Disprove the following statement: For each integer n>1 and each divisor d of φ(n), there is an integer a of order d modulo n.

Any help would be appreciated.

Answer 1

This is in general not true. Note that $\varphi(8)=4$, but there are no elements of order $4$ modulo $8$.

Remark: There is an element of order $\varphi(n)$ modulo $n$ precisely if $n=1$, $2$, $4$, or $p^k$ or $2p^k$ where $p$ is an odd prime.