Suppose $a$, $b$, and $c$ are positive integers such that $\frac{a}{10}+\frac{b}{21}=\frac{c}{210}.$Which of the following statements are necessarily true?
I. If $\gcd (a,10)=1$ or $\gcd (b,21)=1$ or both, then $\gcd (c,210)=1$.
II. If $\gcd (c,210)=1$, then $\gcd (a,10)=1$ or $\gcd (b,21)=1$ or both.
III. $\gcd (c,210)=1$ if and only if $\gcd (a,10)=1$ and $\gcd (b,21)=1$.