Question 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 $\gcd(a,14)=\gcd(b,15)=1.$
$$
\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.
$$
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 $\gcd(a,14)=\gcd(b,15)=1.$