Suppose $a$, $b$, and $c$ are positive integers such that $\frac{a}{6}+\frac{b}{10}=\frac{c}{30}.$Which of the following statements are necessarily true?
I. If $\gcd (a,6)=1$ or $\gcd (b,10)=1$ or both, then $\gcd (c,30)=1$.
II. If $\gcd (c,30)=1$, then $\gcd (a,6)=1$ or $\gcd (b,10)=1$ or both.
III. $\gcd (c,30)=1$ if and only if $\gcd (a,6)=1$ and $\gcd (b,10)=1$.