Let $G$ be a finite group with $|G| = 70$, suppose $a \in G$ satisfies $a^{21} = e$. Prove that $a^7 = e$. Which of the following proofs is wrong?

Question

Let $G$ be a finite group with $|G| = 70$, suppose $a \in G$ satisfies $a^{21} = e$. Prove that $a^7 = e$. Which of the following proofs is wrong?

I got really confused here. Any help would be appreciated.

Answer 1

Proof C is invalid. You can only conclude that the order of $a$ divides $\gcd(70, 21) = 7.$ So we could have $a=e,$ which has order $1 \ne 7.$