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