The product of integers 240 and k is a perfect cube. What is the smallest possible positive value of k?
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I think it's 60
240x60=14400, which is not a perfect cube
240 0= 24 * 10 = 2^3 * 3 * 10 = 2^4 * 3 * 5
We need 2 2s, 2 3s, and 2 5s. This gives us:
2^2 * 3^2 * 5^2 = 900.
niceeee.
here's a way
\(240=2^4\cdot3\cdot5=2^3(2\cdot3\cdot5)\). For \(240k\) to be a perfect cube (and not a perfect square), \(k\) must be at least \(2^2\cdot3^2\cdot5^2=\boxed{900}\).
+44 
