Multiply 210125 by the smallest number so that the product is a perfect cube.

On factorising 210125 into prime factors, we get:

$210125=5 \times 5 \times 5 \times 41 \times 41$

On grouping the factors in triples of equal factors, we get:

$210125=\{5 \times 5 \times 5\} \times 41 \times 41$

It is evident that the prime factors of 210125 cannot be grouped into triples of equal factors such that no factor is left over. Therefore, 210125 is not a perfect cube. However, if the number is multiplied by 41, the factors can be grouped into triples of equal factors such that no factor is left over.

Hence, the number 210125 should be multiplied by 41 to make it a perfect cube.

Also, the product is given as:

$210125 \times 41=\{5 \times 5 \times 5\} \times\{41 \times 41 \times 41\}$

$\Rightarrow 8615125=\{5 \times 5 \times 5\} \times\{41 \times 41 \times 41\}$

To get the cube root of the produce 8615125 , take one factor from each triple. The cube root is $5 \times 41=205$.

Hence, the required numbers are 41 and 205.