{"status": "success", "data": {"description_md": "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 $\\gcd(a,14)=\\gcd(b,15)=1$.\n\n$\\textbf{(A)}~\\text{I, II, and III}\\qquad\\textbf{(B)}~\\text{I only}\\qquad\\textbf{(C)}~\\text{I and II only}\\qquad\\textbf{(D)}~\\text{III only}\\qquad\\textbf{(E)}~\\text{II and III only}$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

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

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

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2023 AMC 12B Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12B_p16", "prev": "/problem/23_amc12B_p14"}}