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Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We prove thatfor any positive integer $k$, there is a number $m$ for which the equation$\sigma(x)=m$ has exactly $k$ solutions, settling a conjecture of Sierpi\'nskifrom 1955. Additionally, it is shown that for every positive even $k$, there isa number $m$ for which the equation $\phi(x)=m$ has exactly $k$ solutions,where $\phi$ is Euler's function, making progress toward another conjecture ofSierpi\'nski from 1955.

On two conjectures of Sierpi\'nski concerning the arithmetic functions $\sigma$ and $\phi$