About primitive roots of $p$ less than $\frac{p-1}2$.

Question

If all the residues of $p$, from $2$ to $\dfrac{p-1}2$ can be expressed as $(2^A)(3^B)(5^C) \pmod p$ for some integers $A$, $B$, $C$ then is it true at least one of $2$ or $3$ or $5$ is a primitive root$\pmod p$?