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How many positive perfect cube factors does 3*6*5*10*12 have?

 Dec 8, 2021
 #1
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How many positive perfect cube factors does 3*6*5*10*12 have?     

 

Break down each of the listed factors to its prime factors.  

 

3     •     2 • 3     •     5     •     2 • 5     •     2 • 2 • 3    

 

Now group like terms.  

 

2 • 2 • 2 • 2 • 3 • 3 • 3 • 5 • 5   

 

I see two cubes in there, namely, 23 and 33   

 

So, I see two perfect cubes.  

.

 Dec 8, 2021
 #2
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(1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, 36, 40, 45, 48, 50, 54, 60, 72, 75, 80, 90, 100, 108, 120, 135, 144, 150, 180, 200, 216, 225, 240, 270, 300, 360, 400, 432, 450, 540, 600, 675, 720, 900, 1080, 1200, 1350, 1800, 2160, 2700, 3600, 5400, 10800)>>Total = 60

 

There are 4 prfect cubes: 1, 8, 27, 216

10800 = 2^4 * 3^3 * 5^2 ==Floor[4/3] + 1  *   Floor[3/3] + 1 ==2 x 2 ==4 perfect cubes.

 Dec 8, 2021

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