Question

Which of the following are perfect cubes ?

v. $1728$

Answer 1

Step: Check if $1728$ is a perfect cube

Prime factorization of $1728$:

$\begin{array}{cc}2& 1728\\ 2& 864\\ 2& 432\\ 2& 216\\ 2& 108\\ 2& 54\\ 3& 27\\ 3& 9\\ 3& 3\\ & 1\end{array}$

$\Rightarrow 1728=\{2\times 2\times 2\}\times \{2\times 2\times 2\}\times \{3\times 3\times 3\}$

Clearly, the prime factors of $1728$ can be grouped in triplets of equal factors and no factor is left over.

Hence, $1728$ is a perfect cube.