I am confused by a following problem.
This is well-known for using in a proof of Artin reciprocity law, so I believe someone remember this.
Problem. Let $n=q_1^{r_1}\cdots q_s^{r_s}$ be the factorization of the integer $n$ as a product distinct primes and let $a>1$ be an integer.There exists infinitely many square-free integers $m=p_1\cdots p_sp_1'\cdots p_s'$ such that the order of $a\ \rm{mod}\ m$ is divisible by $n$.
Please tell me a proof of this or answer to Why as $r$ increases, prime $p$ s.t. $\operatorname{ord} a=q^r$ (in mod $p$) also increases?.
But you can use the fact in the link above.
This fact also appears in Serge Lang "Algebraic Number Fields" p.201.
Edit Nancy Childress "Class Field Theory" p.116 avoides this trouble to prove Artin's lemma.