The smallest number by which $(4{)}^2\times \sqrt[3]{31000}$ must be divided to make it a perfect cube is _______.

The correct option is D: $20$

$(4{)}^2\times \sqrt[3]{31000}=4\times 4\times \sqrt[3]{310\times 10\times 10}=4\times 4\times 10$

$4\times 4\times 10=2\times 2\times 2\times 2\times 2\times 5=\underset{–}{(2\times 2\times 2)}\times 2\times 2\times 5$

So, the given number mute divided by $2\times 2\times 5=20$ to make a perfect cube.