Factoring and Divisibility Problems with 2^n - 1: Beginner Proof Method

[Hivoyer]

Homework Statement

I'm given this problem and I think I'm supposed to use the same or similar method to solve both of its parts:

a) Factor $$2^{15} - 1 = 32,767$$ into a product of two smaller positive integers.
b) Find an integer $$x$$ such that $$1 < x < 2^{32767} - 1$$ and $$2^{32767}$$ is divisible by $$x$$.

Homework Equations

It is shown above the problem that:
$$x = 1 * 2 * 3 * 4 * ... * (n + 1) + 2 = 2 * (1 * 3 * 4 * ... *(n + 1) + 1$$
While I get that it's true, I don't quite see how I can apply the same to solving the problem.Can anyone give a hint?

The Attempt at a Solution

I tried "guessing", however with no success.

 

[haruspex]

I don't see any connection between the two.
Can you factorise x³-1?

 

[HallsofIvy]

For b, you realize that $2^{32767}$ can be divided evenly only by another power of 2, right? So x must be a power of 2.