Latest CAT Aptitude Question SOLUTION: The number of positive integer form 1 to 10^6(both inclusive )which are neither perfect squares ,nor cubes,nor perfect fourth powers is:

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The number of positive integer form 1 to 10^6(both inclusive )which are neither perfect squares ,nor cubes,nor perfect fourth powers is:

Asked In CAT Sakshi (10 years ago)

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Veena V  (10 years ago)

There are 1000 perfect squares between 1 and 1000000 (1000^2 = 1000000). This list also includes 31 perfect 4th powers (31^4 = 923521, the largest perfect 4th power below 1000000).
There are 100 perfect cubes between 1 and 1000000 (100^3 = 1000000). However, this list of cubes also includes integers that are also squares - the largest of them being 1000000 (10^2^3 = 1000000). Therefore, this list of cubes includes 10 integers which are squares as well: 1^2^3, 2^2^3, 3^2^3, 4^2^3, 5^2^3, 6^2^3, 7^2^3, 8^2^3, 9^2^3 and 10^2^3.
So the total number of integers that are perfect squares, perfect cubes or perfect 4th powers between 1 and 1000000 is 1000 + 100 - 10 = 1090.
The total number of integers that are neither perfect squares, nor perfect cubes nor perfect 4th powers between 1 and 1000000 is 1000000 - 1090 = 998910

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