Find the smallest $4$ digit number which is a factor of $2005^6 - 1$. Find the smallest $4$ digit number which is a factor of $2005^6 - 1$.
My attempt:
We see that $(2005-1)$ is a factor of $2005^6-1$ by the factorization of $2005^6-1$. Therefore, $1002$ is a factor of $2005^6-1$. I then have to show that $1001$ is not a factor but how do I do that? Should I use the factorization of $x^6-1$?

Let's try basic algebra to figure this out:
$$ (2,005^6 = 64,966,020,037,537,515,625) - 1 = 64,966,020,037,537,515,624 $$
We already know that this number isn't divisible by $1,000$, so what about the others?
$$ \frac{2,005^6 - 1}{1,001} = 64,901,118,918,618,896.72 $$
$$ \frac{2,005^6 - 1}{1,002} = 64,836,347,342,851,812 $$
The real answer to this is $1,002$ then.
