+0  
 
+1
42
4
avatar+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
avatar+370 
-1

I think it's 60

 Jul 26, 2022
 #2
avatar+1118 
+4

240x60=14400, which is not a perfect cube

nerdiest  Jul 26, 2022
 #3
avatar+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
avatar+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

17 Online Users