1. **Problem statement:** Find the smallest positive integer $n$ such that $n + 2035$ is a perfect cube. 2. **Understanding the problem:** We want to find $n > 0$ so that $n + 2035 = k^3$ for some integer $k$. 3. **Approach:** - First, find the cube root of 2035 to estimate $k$. - Then check cubes of integers greater than this estimate until $k^3 - 2035$ is positive. 4. **Calculate approximate cube root:** $$\sqrt[3]{2035} \approx 12.67$$ 5. **Check cubes starting from $k=13$:** - $13^3 = 2197$ - Calculate $n = 2197 - 2035 = 162$ 6. **Check if $n$ is positive:** - $162 > 0$, so $n=162$ is a valid solution. 7. **Verify smaller $n$ values:** - For $k=12$, $12^3=1728$, $1728-2035=-307$ (negative, invalid) 8. **Conclusion:** The smallest positive integer $n$ is $162$. **Final answer:** $$n = 162$$