2023 AMC 10B Problem 18

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)} \mathrm{I}, \mathrm{II}, and \mathrm{III} \qquad \textbf{(B)} \mathrm{I} only \qquad \textbf{(C)} \mathrm{I} and \mathrm{II} only \qquad \textbf{(D)} \mathrm{III} only \qquad \textbf{(E)} \mathrm{II} and \mathrm{III} only

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.