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What is the smallest positive integer n such that the fourth root of 56*n*360 is an integer?

 Apr 27, 2023
 #1
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What is the smallest positive integer n such that the fourth root of 56*n*360 is an integer?  

 

 

I don't know if it's the smallest integer, but I can help with an answer.  Here's what I did:  

 

I broke down 56 and 360 into their smallest factors 

 

36 = 2, 2, 2, and 7    and    360 = 2, 2, 2, 3, 3, and 5  

 

Combined those and so    56 * 360 = 2, 2, 2, 7, 2, 2, 2, 3, 3, and 5    

 

We want a perfect 4th power, so I added enough of each factor to make 4 of them   

 

The ones I added are

highlighted in dark blue               2, 2, 2, 2, 7, 7, 7, 7, 2, 2, 2, 2, 3, 3, 3, 3, 5, 5, 5, and 5  

 

Multiply the added (blue) ones together    2 * 7 * 7 * 7 * 2 * 3 * 3 * 5 * 5 * 5  =  1,543,500  

 

So 1,543,500 is one integer that will work.  I just don't know if it's the smallest. 

 

 

Check answer:   56 * 360 * 1,543,500  =  31,116,960,000   the 4th root of which is 420  

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 Apr 27, 2023
 #2
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ooops, I made a typo  

 

36 = 2, 2, 2, and 7  should say  56 = 2, 2, 2, and 7  

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 Apr 27, 2023
 #3
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The final answer is 420.

 

eggy car

 Apr 27, 2023
 #4
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Let's test 420 ... 56 * 420 * 360 = 8,467,200   

 

Now taking the 4th root of 8,467,200 ...... 

 

The 4th root is 53.94298 ... NOT AN INTEGER  

 

So 420 is wrong    

Guest Apr 27, 2023

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