Which two numbers are both square and cube numbers ?
1
1 and 8
2
8 and 27
3
27 and 64
4
1 and 64
Official Solution
Correct Option: (4)
To determine which numbers are both square and cube numbers, we must identify numbers that satisfy being both perfect squares and perfect cubes.
A number is a square number if it can be expressed as where is an integer.
A number is a cube number if it can be expressed as where is an integer.
A number that is both a square and a cube must be a perfect sixth power, i.e., . We will analyze each option to check the numbers that are perfect sixth powers:
1: 1 can be both and , hence it is a sixth power (i.e., ).
8: 8 can be expressed as but not as a perfect square, so it is not a sixth power.
27: 27 can be expressed as but not as a perfect square, so it is not a sixth power.
64: 64 can be expressed as and also as , thus it is a sixth power (i.e., ).
Thus, the numbers "1" and "64" are both square and cube numbers. Therefore, the correct answer is the combination that includes these numbers: