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A number \(x\) is equal to\(7\cdot24\cdot48\) . What is the smallest positive integer \(y\) such that the product \(xy\) is a perfect cube?

 Mar 27, 2020
 #1
avatar+1970 
+1

Sorry I'm not CPhill, but here:

 

7*24*48=8064 =2^7*9*7

 

Therefore, y =2^2*3*7^2 =588

=8064*588=47416

 

Recheck: 168^3=47416

 

Hope this helped!

 Mar 27, 2020
edited by CalTheGreat  Mar 27, 2020
 #2
avatar+111330 
+1

x  =  7 * 24 * 48  = 8064

 

Factor  24  =   3 * 2^3

 

Factor  48  =  3 *  2^4

 

So

 

x  =  7 * 3 * 3 * 2^3 * 2^4   =   7 * 3^2  * 2^7

 

Note  that for xy to be a perfect  cube, y  needs to be   7^2 * 3 * 2^2   =  588

 

 

So....to check

 

xy =  8064 * 588  =  4741362

 

Cube root    = 168

 

 

cool cool cool

 Mar 27, 2020
 #3
avatar+1970 
+1

Yay I got the same answer as Chris!

CalTheGreat  Mar 27, 2020

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