List all elements of $\displaystyle Z_{40}$ with order 10.
My solution: By a theorem, I know that the unique subgroup of order 10 is <40/10> = <4> = {0,4,8,12,16,20,24,28,32,36>
So are those the answers?
Suppose $\displaystyle k$ is an element of $\displaystyle Z_{40}$ of order $\displaystyle 10$, then:
$\displaystyle
10k \equiv 0 \mod 40
$
or there exists a $\displaystyle \lambda \in \bold{N}$ such that:
$\displaystyle
10k=\lambda 40
$
which imples that $\displaystyle k$ is a multiple of $\displaystyle 4$.
But not all multiples of $\displaystyle 4$ are of order $\displaystyle 10$, for example $\displaystyle 8$ is of order $\displaystyle 5$.
RonL