Question

Which two numbers are both square and cube numbers ?

• A. 1 and 8
• B. 8 and 27
• C. 27 and 64
• D. 1 and 64

Answer 1

The Correct Option is D

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 \( n^2 \) where \( n \) is an integer.
• A number is a cube number if it can be expressed as \( n^3 \) where \( n \) is an integer.

A number that is both a square and a cube must be a perfect sixth power, i.e., \( n^6 \). We will analyze each option to check the numbers that are perfect sixth powers:

• 1: 1 can be both \(1^2\) and \(1^3\), hence it is a sixth power (i.e., \(1^6\)).
• 8: 8 can be expressed as \(2^3\) but not as a perfect square, so it is not a sixth power.
• 27: 27 can be expressed as \(3^3\) but not as a perfect square, so it is not a sixth power.
• 64: 64 can be expressed as \(8^2\) and also as \(4^3\), thus it is a sixth power (i.e., \(2^6\)).

Thus, the numbers "1" and "64" are both square and cube numbers. Therefore, the correct answer is the combination that includes these numbers:

1

64