  We propose a generic algorithm for computing the inverses of a multiplicative
function under the assumption that the set of inverses is finite. More
generally, our algorithm can compute certain functions of the inverses, such as
their power sums (e.g., cardinality) or extrema, without direct enumeration of
the inverses. We illustrate our algorithm with Euler's totient function
$\varphi(\cdot)$ and the $k$-th power sum of divisors $\sigma_k(\cdot)$. For
example, we can establish that the number of solutions to $\sigma_1(x) =
10^{1000}$ is 15,512,215,160,488,452,125,793,724,066,873,737,608,071,476, while
it is intractable to iterate over the actual solutions.
