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# Help me

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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

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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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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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