Let f be a function from $\{1, 2, . . . , 22\}$ to the positive integers such that $mn | f(m) + f(n)$ for all $m, n \in \{1, 2, . . . , 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$.

Let's think step by step and output the final answer within \boxed{}.
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