Write the cubes of $7$ natural numbers which are of the form $3n+2(\mathrm{i}.\mathrm{e}.,5,8,11,14,...)$ and verify that " The cube of the natural number of the form $3n+2$ when divided by $3$ leaves remainder $2$".

Find the smallest number by which the following numbers must be multiplied so that the products are perfect cubes.
$5324$

Find the smallest number by which the following numbers must be multiplied so that the products are perfect cubes.
$1323$

Find the smallest number by which the following numbers must be multiplied so that the products are perfect cubes.
$27783$

Find the smallest number by which the following numbers must be multiplied so that the products are perfect cubes.
$12,348$

Find the smallest number by which the following numbers must be multiplied so that the products are perfect cubes.
$3125$

Find the smallest number by which the following numbers must be multiplied so that the products are perfect cubes.
$101306$

Find the smallest number by which the following should be divided to get the quotient as a perfect cube.

$1,536$