{"status": "success", "data": {"description_md": "Suppose $a$, $b$, and $c$ are positive integers such that\n$$\\dfrac{a}{14}+\\dfrac{b}{15}=\\dfrac{c}{210}.$$\nWhich of the following statements are necessarily true?\nI. If $\\gcd(a,14)=1$ or $\\gcd(b,15)=1$ or both, then $\\gcd(c,210)=1$.\nII. If $\\gcd(c,210)=1$, then $\\gcd(a,14)=1$ or $\\gcd(b,15)=1$ or both.\nIII. $\\gcd(c,210)=1$ if and only if $\\gcd(a,14)=\\gcd(b,15)=1$.\n\n$\\textbf{(A)}~\\text{I, II, and III}\\qquad\\textbf{(B)}~\\text{I only}\\qquad\\textbf{(C)}~\\text{I and II only}\\qquad\\textbf{(D)}~\\text{III only}\\qquad\\textbf{(E)}~\\text{II and III only}$", "description_html": "<p>Suppose  <span class=\"katex--inline\">a</span> ,  <span class=\"katex--inline\">b</span> , and  <span class=\"katex--inline\">c</span>  are positive integers such that<br/>\n <span class=\"katex--display\">\\dfrac{a}{14}+\\dfrac{b}{15}=\\dfrac{c}{210}.</span> <br/>\nWhich of the following statements are necessarily true?<br/>\nI. If  <span class=\"katex--inline\">\\gcd(a,14)=1</span>  or  <span class=\"katex--inline\">\\gcd(b,15)=1</span>  or both, then  <span class=\"katex--inline\">\\gcd(c,210)=1</span> .<br/>\nII. If  <span class=\"katex--inline\">\\gcd(c,210)=1</span> , then  <span class=\"katex--inline\">\\gcd(a,14)=1</span>  or  <span class=\"katex--inline\">\\gcd(b,15)=1</span>  or both.<br/>\nIII.  <span class=\"katex--inline\">\\gcd(c,210)=1</span>  if and only if  <span class=\"katex--inline\">\\gcd(a,14)=\\gcd(b,15)=1</span> .</p>\n<p> <span class=\"katex--inline\">\\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}</span> </p>\n<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2023 AMC 10B Problem 18", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc10B_p19", "prev": "/problem/23_amc10B_p17"}}