Suppose $a$, $b$, and $c$ are positive integers such that

$\frac{a}{22}+\frac{b}{35}=\frac{c}{770}$.

Which of the following statements are necessarily true?

• I. If $\gcd (a,22)=1$ or $\gcd (b,35)=1$ or both, then $\gcd (c,770)=1$.
• II. If $\gcd (c,770)=1$, then $\gcd (a,22)=1$ or $\gcd (b,35)=1$ or both.
• III. $\gcd (c,770)=1$ if and only if $\gcd (a,22)=\gcd (b,35)=1$.