{"status": "success", "data": {"description_md": "The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer?\n\n$\\textbf{(A)}\\ 17\\qquad\\textbf{(B)}\\ 18\\qquad\\textbf{(C)}\\ 19\\qquad\\textbf{(D)}\\ 20\\qquad\\textbf{(E)}\\ 21$\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>The sequence  <span class=\"katex--inline\">(a_n)</span>  is defined recursively by  <span class=\"katex--inline\">a_0=1</span> ,  <span class=\"katex--inline\">a_1=\\sqrt[19]{2}</span> , and  <span class=\"katex--inline\">a_n=a_{n-1}a_{n-2}^2</span>  for  <span class=\"katex--inline\">n\\geq 2</span> . What is the smallest positive integer  <span class=\"katex--inline\">k</span>  such that the product  <span class=\"katex--inline\">a_1a_2\\cdots a_k</span>  is an integer?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 17\\qquad\\textbf{(B)}\\ 18\\qquad\\textbf{(C)}\\ 19\\qquad\\textbf{(D)}\\ 20\\qquad\\textbf{(E)}\\ 21</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": "2016 AMC 12B Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/16_amc12B_p24"}}