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Help how to do this

 

Cai writes down the list of positive integers, excluding squares and cubes and all perfect powers.  His sequence starts
2, 3, 5, 6, 7, 10, 11, ...
What is the 100th term in Cai's list?

 Apr 16, 2023
 #1
avatar+144 
+5

The answer is 97.

If you wanted the explanation too, please let me know! (I didn't really want to write down all of my calculations, as I am admittedly a bit lazy :P)

Hope this helped!

 Apr 16, 2023
 #2
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How can the 100th term be 97 when at least half a dozen perfect squres, perfect qubes and perfect powers are excluded between 1 and 100 ?! Which means that the 100th term is greater than 100.

 

To the OP: Don't be lazy! List all the numbers between 1 and 100 and remove all perfect squares, perfect cubes and perfect powers and count what it remains. If you don't know what a perfect square or perfect cube or perfect power is, try this on your calculator: 1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25......10^2=100 and so on. 1, 4, 9, 16, 25......100 are perfect squares.

 

Do the same thing with perfect cubes: 1^3 =1,  2^3=8, 3^3=27, 4^3=64.......and so on. 1, 8, 27, 64.....are perfect cubes.

 

For perfect powers, generally all perfect squares and perfect cubes are perfect powers as well. However: 2^3=8, 2^4=16, 2^5 =32, 2^6=64........etc. Then: 3^1=3, 3^2=9, 3^3=27, 3^4=81....and then try 4^1=4, 4^2=16, 4^3=64......and so on.....Then try 5, 6, 7, 8, 9, 10......etc. 

Guest Apr 17, 2023

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