You know that a perfect cube number is the product of three equal real numbers.

\[49000={{2}^{3}}\times {{5}^{3}}\times {{7}^{2}}\]

By inspection, you notice that 7 is the minimum positive integer to be multiplied with "49000" to obtain your required perfect cube in the following way.

\[\begin{align}

  & 49000\times 7={{2}^{3}}\times {{5}^{3}}\times {{7}^{2}} \\ 

 & 49000\times 7={{2}^{3}}\times {{5}^{3}}\times {{7}^{2}}\times 7 \\ 

 & 343000={{\left( 2\times 5\times 7 \right)}^{3}} \\ 

 & 343000={{\left( 70 \right)}^{3}}  

\end{align}\]

The required minimum number is 7