Suppose $a$, $b$, and $c$ are positive integers such that $\frac{a}{12}+\frac{b}{10}=\frac{c}{60}.$Which of the following statements are necessarily true?
I. If $\gcd (a,12)=1$ or $\gcd (b,10)=1$ or both, then $\gcd (c,60)=1$.
II. If $\gcd (c,60)=1$, then $\gcd (a,12)=1$ or $\gcd (b,10)=1$ or both.
III. $\gcd (c,60)=1$ if and only if $\gcd (a,12)=1$ and $\gcd (b,10)=1$.