Find the smallest $4$ digit number which is a factor of $2005^6 - 1$.

Question

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 $x^6-1$?

Answer 1

$2005^6-1\equiv 3^6-1\not \equiv 0 \pmod {1001}$. Further, as mentioned in comments by Steven, you need to eliminate $1000$ as a factor too.

Answer 2

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.