{"status": "success", "data": {"description_md": "Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that $$\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}=\\frac{h_n}{k_n}.$$ Let $L_n$ denote the least common multiple of the numbers $1, 2, 3, \\ldots, n$. For how many integers with $1\\le{n}\\le{22}$ is $k_n<L_n$?\n\n$\\textbf{(A) }0 \\qquad\\textbf{(B) }3 \\qquad\\textbf{(C) }7 \\qquad\\textbf{(D) }8\\qquad\\textbf{(E) }10$\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>Let  <span class=\"katex--inline\">h_n</span>  and  <span class=\"katex--inline\">k_n</span>  be the unique relatively prime positive integers such that  <span class=\"katex--display\">\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}=\\frac{h_n}{k_n}.</span>  Let  <span class=\"katex--inline\">L_n</span>  denote the least common multiple of the numbers  <span class=\"katex--inline\">1, 2, 3, \\ldots, n</span> . For how many integers with  <span class=\"katex--inline\">1\\le{n}\\le{22}</span>  is  <span class=\"katex--inline\">k_n&lt;L_n</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }0 \\qquad\\textbf{(B) }3 \\qquad\\textbf{(C) }7 \\qquad\\textbf{(D) }8\\qquad\\textbf{(E) }10</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": 5, "problem_name": "2022 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/22_amc12A_p24", "prev": "/problem/22_amc12A_p22"}}