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avatar+2856 

What integer n has the property that \(5^{96}\) is greater than \(n^{72}\) and \(5^{96}\) is less than \((n+1)^{72}\).

 

 

 

So I did

 

5^96>n^72

 

5^96<(n+1)^72

 

Then take 12th root

 

5^8>n^6

 

5^8<(n+1)^6

 

Then I take square root

 

5^4>n^3

 

5^4<(n+1)^3

 

Then I easily deduced that n must be 8.

 

Is my solution correct?

 May 24, 2019
 #1
avatar+111438 
+1

Well.....   there's one sure way to know......

 

5^96 > 8^72  ????  True

 

5^96 < (9)^72 ???? True

 

Looks like your solution is just fine, CU!!!

 

Very ingenious !!!!

 

 

cool cool cool

 May 24, 2019
edited by CPhill  May 24, 2019
 #2
avatar+110226 
+1

I am not saying that your answer is wrong CalculatorUser  BUT did you consider the possibility that n could be negative?

 May 25, 2019
 #3
avatar+30661 
+3

n72 < 596 < (n+1)72  

 

n < 596/72 < n+1

 

n < 54/3 < n+1

 

n < 8.55 < n+1

 

n = 8

 May 25, 2019
 #4
avatar+110226 
+1

I know that is right Alan and I think CalculatorUser does too, he wanted to do it without the calc I think.

BUT I just wanted him/her to think about wether there could be a negative answer.

I know the answer to this, I just want to be sure that CalculatorUser has thought about it too.  

Melody  May 25, 2019
edited by Melody  May 25, 2019
 #5
avatar+2856 
+1

Yes thank you, Melody, I never thought that my answer could be negative. I am lucky that the problem only had one answer.

 

Next time in these problems, I will make sure there are no negative answers

 May 25, 2019
 #6
avatar+110226 
+1

I suggest you think very hard about THIS  answer/question

You should not just accept what some answer written down somewhere is. 

 

How do you know there is not a negative answer?  Answer: Someone told you so.

 

Well that is not you knowing, that is at best someone else knowing.

 

If you think about it purely as a logic exercise, you will work out why there cannot be a negative answer.

I took me a while to get my head around it so it might take you a while too but the reason is not very complex.   wink

Melody  May 26, 2019

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