$72=\underline{2\times 2\times 2}\times 3\times 3$

The prime factor 3 does not appear in a group of three. Therefore, 72 is not a perfect cube. To make it a cube, we need one more 3.

In that case, $72\times 3=\underline{2\times 2\times 2}\times \underline{3\times 3\times 3}=216$ is a perfect cube.
Hence, the smallest natural number by which 72 should be multiplied to make it a perfect cube is 3.