Find what values of $n$ give $\varphi(n) = 10$ 
For what values of $n$ do we get $\varphi(n) = 10$?

Here, $\varphi$ is the Euler Totient Function. Now, just by looking at it, I can see that this happens when $n = 11$. Also, my friend told me that it happens when $n = 22$, but both of these are lucky guesses, or educated guesses. I haven't actually worked it out, and I don't know if there are any more.
How would I go about answering this question?

If a number $n$ has prime factorization $p_1^{x_1} p_2^{x_2} p_3^{x_3} \cdots $, then the following formula holds:
$$
\varphi(n) = \left( p_1^{x_1} - p_1^{x_1 - 1}\right)\left( p_2^{x_2} - p_2^{x_2 - 1}\right) \left( p_3^{x_3} - p_3^{x_3 - 1}\right) \cdots
$$
