Question

Find the smallest positive integer n for which 864x n is a perfect cube

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ 864\times \mathrm{n}=108\times 8\times \mathrm{n} \\ =12\times 9\times 2^3\times \mathrm{n} \\ =2^3\times 3^3\times 2^2\times \mathrm{n} \\ \mathrm{So},\mathrm{if}\mathrm{n}=2,\mathrm{then} \\ 864\times 2=2^3\times 3^3\times 2^3 \\ ={\left(2\times 3\times 2\right)}^3 \\ ={12}^3 \\ \mathrm{So}\sqrt[3]{864\times 2}=12 \\ \mathrm{So}\mathrm{minimum}\mathrm{value}\mathrm{of}\mathrm{n}\mathrm{is}2. \\ \mathrm{Regards} \end{gathered}$$