Abstract-several questions

Well, I'm trying to solve some old exams in abstract algebra, and I need your help in the following:

**Question 1:**

Let $\displaystyle U(Z/15Z) $ be the multiplicative group of the invertible elemenets in $\displaystyle Z/15Z $ . Write $\displaystyle U(Z/15Z) $ as a product of cyclic groups.

**My try:**

It's easy to show that x is in U iff gcd(x,15)=1... Hence:

U={1,2,8,4,7,13,11,14} ... If we'll take the element 2 to be a generator we'll get the cyclic group {1,2,4,8} ... If we'll take <7> we'll get {1,7,4,13} etc... I can't figure out how to write U as a product of cyclic groups when the intersection of each two cyclic subgroups of U is "bigger" than {1}...

**Question 2:**

Prove that $\displaystyle Z/nZ $ has only one maximal ideal (which isn't trivial) iff n is a power of a prime number.

**Question 3:**

Find a 3-sylow subgroup of $\displaystyle S_{8} $ and find a group that is isomorphic to it.

**My try:**

We know $\displaystyle o(S_{8}) = 8! $ . Hence, a 3-sylow subgroup H is from order $\displaystyle o(H)= 3^{2}=9 $. If we'll be able to find an element in $\displaystyle S_{8} $ from that order- we're done...But is there any element of that order? How should I solve this one?

Thanks a lot!