Assume that the prime factorization of $M=q_1^{e_1}q_2^{e_2}...q_s^{e_s}$, and $p^n$ divides the order of $a \pmod M$, where $p$ is prime, and $gcd(p^n,M)=1$.
Now, my question is:
How to prove that $p^n$ must divide the order of $a \pmod {q_c}$ for some $q_c$ that divides $M$?
I have a simple proof of the above question, but I don't know if it is right or wrong :
proof, since $ord_M(a)=lcm(ord_{q_1^{e_1}}(a), ord_{q_2^{e_2}}(a),....., ord_{q_s^{e_s}}(a))$ and $p^n$ divides $ord_M(a)$, then, $p^n$ must divide $ord_{q_c^{e_c}}(a)$ for some $q_c$ that divides $M$.
Now, since $ord_{q_c^{e_c}}(a)=q_c^k \times ord_{q_c}(a)$, where $k \leq e_c-1$, and since $gcd(p^n,M)=1$, then, $p^n$ must divide the order of $a \pmod {q_c}$.
if anyone has any idea, please share it.