About primitive roots and ways of expressing them.

Question

If $q$ is a primitive root mod $p$ and $q = (2^a)(3^b)(5^c)\pmod p$ for some $a,b,c$ all elements of integers then say that this $q$ has the 'form' $\{2,3,5\}(p)$. Then is it true all the residues $\pmod p$ have this ${2,3,5}(p)$ form given at least one of them, a primitive root, is in this form?