AMC 12 2023 B Problem 15

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

$$\dfrac{a}{14}+\dfrac{b}{15}=\dfrac{c}{210}$$

Which of the following statements are necessarily true?
I. If \operatorname{gcd}(a, 14)=1 or \operatorname{gcd}(b, 15)=1 or both, then \operatorname{gcd}(c, 210)=1.

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

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

Answer Choices
A. I, II, and III
B. I only
C. I and II only
D. III only
E. II and III only