how many integers between 50000 and 60000 are perfect squares
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Between 1 and 1,000,000 , there are 1000 Perfect Squares and 100 Perfect Cubes.
1000000 = 1000^2 = 100^3
Take any number between 1 and 1000000, it should be of the form -
A^2 and at the same time B^3
i.e. A^2 = B^3
where, A lies between 1 and 1000
B lies between 1 and 100
Now, B^3 will be a Perfect Square only when B itself is a Perfect Square
(Write down the prime factorization for B^3 to understand this)
So, just find the perfect squares between 1 and 100, which are -
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (=B)
Take their cubes -
1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000 (=B^3)
So, these are the numbers which are Perfect Squares as well as Perfect Cubes. And to get your answer, just subtract 10 from the total number of perfect squares.
1000-10 = 990
1000000 = 1000^2 = 100^3
Take any number between 1 and 1000000, it should be of the form -
A^2 and at the same time B^3
i.e. A^2 = B^3
where, A lies between 1 and 1000
B lies between 1 and 100
Now, B^3 will be a Perfect Square only when B itself is a Perfect Square
(Write down the prime factorization for B^3 to understand this)
So, just find the perfect squares between 1 and 100, which are -
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (=B)
Take their cubes -
1, 64, 729, 4096, 15625, 46656, 117649, 262144, 531441, 1000000 (=B^3)
So, these are the numbers which are Perfect Squares as well as Perfect Cubes. And to get your answer, just subtract 10 from the total number of perfect squares.
1000-10 = 990
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