Order of elements in $\Bbb Z_{24}$ What elements of $\Bbb Z_{24}$ are order $2$? Order $3$? Order $4$? Order $6$?
Order $2: 12$
Order $3: 8,16$
Order $4: 6,18$
Order $6: 4$
I had listed element $20$ as order $6$, but erased it late at night. A week later, I can't for the life of me remember why I didn't include it.
Am I correct or incorrect (and why) that element $20$ is order $6$?

You can brute force calculate:
\begin{align*}
20 \cdot 1 \equiv 20 \mod 24 \\
20 \cdot 2 \equiv 16 \mod 24 \\
20 \cdot 3 \equiv 12 \mod 24 \\
20 \cdot 4 \equiv 8 \mod 24 \\
20 \cdot 5 \equiv 4 \mod 24 \\
20 \cdot 6 \equiv 0 \mod 24 \\
\end{align*}
So the first (positive) multiple of $20$ that produces the group identity $0$ is $6$. Therefore that is the order of $20$.
For bigger calculations in the future, you can instead remember this formula:
The order of any element $k \in \Bbb Z_n$ of the cyclic group with $n \in \Bbb N$ elements is
$$
\boxed{\text{ord}(k) = \frac{n}{\gcd(k, n)}}
$$ In this case, $n = 24$ and $k = 20$ and the greatest common divisor of $20$ and $24$ is $\gcd(20, 24) = 4$. So $\text{ord}(k) = \frac{24}{4} = 6$ as expected.
