Let the least possible value of the smallest of 99 cosecutive integers be x and let the number whose cube is the sum be p, then
[tex] \frac{99}{2} (2x+98)=p^3 \\ \\ 99x+4,851=p^3\\ \\ \Rightarrow x=\frac{p^3-4,851}{99}[/tex]
By substitution, we have that [tex]p=33[/tex] and [tex]x=314[/tex].
Therefore, the least possible value of the smallest of 99 consecutive positive integers whose sum is a perfect cube is 314.
