For every three-digit natural number $n$ (where the leading digit of $n$ is nonzero), consider the number $n_0$ obtained by eliminating all zero digits from $n$. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$ for which $n_0$ is a divisor of $n$ and $n_0 \neq n$.

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