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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).

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Welcome, user48390! I take it you're interested in learning how to prove a conjecture using different proof approaches/methods? Do you know of any proofs of your statement, so we don't duplicate what you already might know? – amWhy Nov 6 '12 at 15:19
What does it mean to be a divisor "in more than one way"? – EuYu Nov 6 '12 at 15:34
@EuYu, I think the OP means using more than one approach (to prove it. I'll edit, to clarify. user48390, correct me if I am wrong. – amWhy Nov 6 '12 at 15:40

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).

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