+0

heeeeelllllpppp

+1
42
4
+44

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

heeeeeeeeeeellllllllppppppppp

Jul 26, 2022

#1
+370
-1

I think it's 60

Jul 26, 2022
#2
+1118
+4

240x60=14400, which is not a perfect cube

nerdiest  Jul 26, 2022
#3
+197
+1

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.

Jul 26, 2022
#4
+1118
+4

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 $$2^2\cdot3^2\cdot5^2=\boxed{900}$$.

nerdiest  Jul 26, 2022
edited by nerdiest  Jul 26, 2022