Number Properties Puzzle – Challenge 5

Challenge:

Number 4 is a perfect square number (2 × 2 = 4) and number 8 is a perfect cube number (2 × 2 × 2 = 8). How many positive perfect square numbers less than 2016 are also perfect cubes?

A- 2

B- 3

C- 4

D- 5

E- 10

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Because there aren’t that many perfect squares less than 2016 that is also a perfect cube, let’s look for the smallest perfect square number. This happens to be number 1. It is a perfect square number (1 × 1 = 1) and a perfect cube number (1×1×1=1). To find the next positive perfect square number less than 2015 that is also a perfect cube, let’s take a look at number 2. \(2^{2} = 4\) is a perfect square and \(2^{3} = 8\) is a perfect cube number and \((2^{2})^{3} = 2^{6} = 64\), which is both a perfect square and cube number. Next perfect square and cube number with the base of 2 is \((2^{6})^{2} =2^{12} = 4096\), which is greater than 2016.
With the same method, \(3^{6} = 729\) is the next perfect square number and cube. \(4^{6} = 4096\) is a perfect square and cube number, but is bigger than 2016. Therefore, numbers 1, 64 and 729 are the perfect square and cube numbers less than 2016.