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## Homework Statement

a) Show that there is exactly one maximal ideal in Z_8 and in Z_9.

b) Show that Z_10 and Z_15 have more than one maximal ideal.

## Homework Equations

I know a maximal ideal is one that is not contained within any other ideal (except for the ring itself)

By Theorem, we know that

*In a commutative ring R with identity, every maximal ideal is prime*.

## The Attempt at a Solution

For

*a)*I was thinking I would just show that all of the classes were subsets of the other classes. i.e. [8/0] is contained in [4] is contained in [2], and [6] is contained in [3] and [2], [9] is contained in [3]. Does that make sense? But I couldn't figure out what to do with [5] and [7]. It seems to me like BOTH of those are maximal ideals, but I'm supposed to prove that there's only one. Also, not quite sure how to formalize this into a proof.

I'm pretty confident on what to do for

*b)*. I just have to show that there's more than one, right? And both Z_7 and Z_9 should be ideals in Z_10 and Z_15, aren't they?

Thanks!