Perfect Squares and Perfect Cubes

An integer greater than 1 is a perfect square if all of the indices of its prime factors are even numbers (multiples of \(2\)).

For example, \(571536\) is a perfect square, since \(571536 = 2^4 \times 3^6 \times 7^2 = (2^2 \times 3^3 \times 7)^2\).

An integer greater than 1 is a perfect cube if all of the indices of its prime factors are multiples of \(3\).

For example, \(10648000\) is a perfect cube, since \(10648000 = 2^6 \times 5^3 \times 11^3 = (2^2 \times 5 \times 11)^3\).

Note:

The square root of a perfect square is a whole number; the cube root of a perfect cube is a whole number.

Also, \(0=0^2=0^3\) and \(1=1^2=1^3\) are both perfect squares and perfect cubes, although the test above is stated for integers greater than 1.