Square cubes
Number Theory
Level
2
A positive integer less than 1000 is not only a perfect square but also a perfect cube.
How many such numbers are there?
The answer is 3.
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1 solution
For a number to be a square, every exponent on its prime decomposition must be even.
For a number to be a cube, every exponent on its prime decomposition must be a multiple of 3.
Therefore, for a number to be a square and a cube at the same time, every exponent on its prime decomposition must be a multiple of 2 and 3, that is, a multiple of 6.
Now, the only primes raised to a multiple of six resulting in a number below 1000 are 2 6 and 3 6 . We have two cases. If we add 1 , wich is a perfect square and a perfect cube also ( 1 = 1 2 = 1 3 ) we have the only three cases we are looking for.
Hence the answer is 3 .