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Find the number of integers from 1 to 1 million that are not perfect squares or perfect cubes.

 May 26, 2020
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The square root of 1,000,000 is 1000; therefore, there will be 1000 perfect squares:

12  =  1

22  =  4

32  =  9

...

9982  =  996,004

9992  =  998,001

10002  =  1,000,000

 

Now, we have to cross-check that with a list of perfect cubes.

Since the cube root of 1,000,000 is 100, there will be 100 numbers on this list.

13  =  1                                 Cross this one off

23  =  8

33  =  27

43  =  64                              Cross this one off  

...

983  =  941,192

993  =  970,299

1003  =  1,000,000               Cross this one off

 

So, there will be a total of 1000 from the list of perfect squares plus all those on the list of perfect cubes that 

aren't crossed off.

 

Then, subtract that total from 1,000,000.

 May 26, 2020

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