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

$\frac{a}{20}+\frac{b}{21}=\frac{c}{420}$.

Which of the following statements are necessarily true?

• I. If $\gcd (a,20)=1$ or $\gcd (b,21)=1$ or both, then $\gcd (c,420)=1$.
• II. If $\gcd (c,420)=1$, then $\gcd (a,20)=1$ or $\gcd (b,21)=1$ or both.
• III. $\gcd (c,420)=1$ if and only if $\gcd (a,20)=\gcd (b,21)=1$.