what is the smallest positive number P such that 10500P is a perfect cube?

Sorry Melody, it's 882.

What is the smallest positive number P such that 10500P is a perfect cube?

10500=105*100=7*3*10^2*5

The smallest possible number is\\

7^2*3^2*2

=49*9*2

$${\mathtt{49}}{\mathtt{\,\times\,}}{\mathtt{9}}{\mathtt{\,\times\,}}{\mathtt{2}} = {\mathtt{882}}$$

the smallest number p can be is 882             Corrections made - Thanks anon :)