Show $\Bbb Z_n$ has even number of subgroups iff $n$ is not a square. I am having trouble proving the following:
Show that $\Bbb Z_n$ has an even number of different subgroups if and only if $n$ is not a square number.
$\Bbb Z_n$ is supposed to be integers modulo $n$ for some $n$ under addition.
 A: Hint: start with a specific example.  This will not directly give you a proof, but it should give you a better understanding of the problem.  Try this:
(1) Take $G={\Bbb Z}_{12}$, that is, $n=12$.  As $G$ is cyclic, all of its subgroups are cyclic, so they are $\langle1\rangle$, $\langle2\rangle$, . . . $\langle12\rangle$.  You could write $\langle0\rangle$ for the last one, but for the purpose of this exercise $\langle12\rangle$ is better.  Write out all elements of every subgroup.  However, not all of these subgroups are different: cross out any duplicates.  Now in every set that remains, write down the smallest element.  What do you notice?
(2) Now do all the same things for ${\Bbb Z}_{16}$.  Do you see any relevant difference between this and the previous case?
Beginning group theory is always difficult, but see how you go.  Good luck!
A: Do you know the subgroups of $\Bbb Z_n$ are those of the form $\langle d\rangle=\{0,d,2d,\ldots\}\simeq \Bbb Z_{n/d}$ for $d\mid n$? Do you know how to count the divisors of a given number $n$?
