AMC 12 2023b Problem 15

Suppose $a$, $b$, and $c$ are positive integers such that$$\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.$$Which of the following statements are necessarily true?

I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$.

II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both.

III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$.

\textbf{(A)}\text{I, II, and III} \textbf{(B)}\text{I only} \textbf{(C)}\text{I and II only} \textbf{(D)}\text{III only} \textbf{(E)}~\text{II and III only}