$240=2^4\cdot3\cdot5=2^3(2\cdot3\cdot5)$. For $240k$ to be a perfect cube (and not a perfect square), $k$ must be at least $2^2\cdot3^2\cdot5^2=\boxed{900}$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The product of integers 240 and $k$ is a perfect cube. What is the smallest possible positive value of $k$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
900
The product of integers 240 and $k$ is a perfect cube. What is the smallest possible positive value of $k$?
test