{"status": "success", "data": {"description_md": "For all positive integers $n$ less than $2002$, let\n\n  \n![\\begin{eqnarray*} a_n =\\left\\{ \\begin{array}{lr} 11, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }13\\ \\text{and\\ }14;\\\\ 13, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }14\\ \\text{and\\ }11;\\\\ 14, & \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }11\\ \\text{and\\ }13;\\\\ 0, & \\text{otherwise}. \\end{array} \\right. \\end{eqnarray*}](https://latex.artofproblemsolving.com/8/d/4/8d49dd65ded465f1090c6122d420d37cdc37d8b0.png)\n\nCalculate $\\sum_{n=1}^{2001} a_n$.\n$\\mathrm{(A)}\\ 448  \\qquad\\mathrm{(B)}\\ 486\\qquad\\mathrm{(C)}\\ 1560\\qquad\\mathrm{(D)}\\ 2001\\qquad\\mathrm{(E)}\\ 2002$\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>For all positive integers <span class=\"katex--inline\">n</span> less than <span class=\"katex--inline\">2002</span>, let</p>&#10;<p><img src=\"https://latex.artofproblemsolving.com/8/d/4/8d49dd65ded465f1090c6122d420d37cdc37d8b0.png\" alt=\"\\begin{eqnarray*} a_n =\\left{ \\begin{array}{lr} 11, &amp; \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }13\\ \\text{and\\ }14;\\ 13, &amp; \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }14\\ \\text{and\\ }11;\\ 14, &amp; \\text{if\\ }n\\ \\text{is\\ divisible\\ by\\ }11\\ \\text{and\\ }13;\\ 0, &amp; \\text{otherwise}. \\end{array} \\right. \\end{eqnarray*}\"/></p>&#10;<p>Calculate <span class=\"katex--inline\">\\sum_{n=1}^{2001} a_n</span>.<br/>&#10;<span class=\"katex--inline\">\\mathrm{(A)}\\ 448  \\qquad\\mathrm{(B)}\\ 486\\qquad\\mathrm{(C)}\\ 1560\\qquad\\mathrm{(D)}\\ 2001\\qquad\\mathrm{(E)}\\ 2002</span></p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2002 AMC 12B Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/02_amc12B_p22", "prev": "/problem/02_amc12B_p20"}}