{"status": "success", "data": {"description_md": "There are $10$ horses, named Horse 1, Horse 2, $\\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let  $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of  $T$?\n\n$\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6$\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": "<p>There are  <span class=\"katex--inline\">10</span>  horses, named Horse 1, Horse 2,  <span class=\"katex--inline\">\\ldots</span> , Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse  <span class=\"katex--inline\">k</span>  runs one lap in exactly  <span class=\"katex--inline\">k</span>  minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time  <span class=\"katex--inline\">S &gt; 0</span> , in minutes, at which all  <span class=\"katex--inline\">10</span>  horses will again simultaneously be at the starting point is  <span class=\"katex--inline\">S = 2520</span> . Let   <span class=\"katex--inline\">T&gt;0</span>  be the least time, in minutes, such that at least  <span class=\"katex--inline\">5</span>  of the horses are again at the starting point. What is the sum of the digits of   <span class=\"katex--inline\">T</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 2\\qquad\\textbf{(B)}\\ 3\\qquad\\textbf{(C)}\\ 4\\qquad\\textbf{(D)}\\ 5\\qquad\\textbf{(E)}\\ 6</span> </p>&#10;<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": "2017 AMC 12A Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/17_amc12A_p13", "prev": "/problem/17_amc12A_p11"}}