Find the smallest positive integer $n$ such that there exists $r \in (0, \tfrac{1}{1000})$ such that the number $(n+r)^3$ is an integer.
Question:
Find the smallest positive integer $n$ such that there exists $r \in (0, \tfrac{1}{1000})$ such that the number $(n+r)^3$ is an integer.
Answer:
Let's think step by step.

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19
Find the smallest positive integer $n$ such that there exists $r \in (0, \tfrac{1}{1000})$ such that the number $(n+r)^3$ is an integer.
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