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Using a family of mock modular forms constructed by Zagier, we study thecoefficients of a mock modular form of weight $3/2$ on$\operatorname{SL}_2(\mathbb{Z})$ modulo primes $\ell\geq 5$. Thesecoefficients are related to the smallest parts function of Andrews. As anapplication, we reprove a theorem of Garvan regarding the properties of thisfunction modulo $\ell$. As another application, we show that congruences modulo$\ell$ for the smallest parts function are rare in a precise sense.

Congruences for a mock modular form on $\operatorname{SL}_2(\mathbb{Z})$ and the smallest parts function