multiple approaches/ways to prove that $1000^N - 1$ cannot be a divisor of $1978^N - 1$

Question

Am interested in learning to do multiple proofs for the same problem, and hence I chose this problem:

Prove that for any natural number $N$,

$1000^N - 1$ cannot be a divisor of $1978^N - 1$.

I'd like to learn how to prove such a statement in more than one way (approach).

Answer 1

Hint $\ $ Examining their factorizations for small $\rm\,N\,$ shows that the power of $3$ dividing the former exceeds that of the latter (by $2),$ so the former cannot divide the latter. It suffices to prove by induction that this pattern persists (which requires only simple number theory).