Question 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 $\gcd(a, 14) = \gcd(b, 15) = 1.$
\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 $\gcd(a, 14) = \gcd(b, 15) = 1.$