Ok so I'm reading Herstein's Algebra and there is a theorem that states that if $G$ is group of order $pq$ where $p,q$ are primes, $p>q$ and if $q$ doesn't divide $p-1$ then $G$ must be cyclic.
So this problem is basically special case and it is supposed to be proved same as the theorem. The part where I'm stuck at is why does $i^6-1=7k$ and $i^5-1=7m$ together with assumption that $q$ does not divide $p-1$ (in this case $5$ does not divide $7-1$), imply that $i$ is also congruent to $1\pmod{7}$.