{"status": "success", "data": {"description_md": "In base $10$, the number $2013$ ends in the digit $3$.  In base $9$, on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$.  For how many positive integers $b$ does the base-$b$-representation of $2013$ end in the digit $3$?\n\n$\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 9\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 16\\qquad\\textbf{(E)}\\ 18$", "description_html": "<p>In base  <span class=\"katex--inline\">10</span> , the number  <span class=\"katex--inline\">2013</span>  ends in the digit  <span class=\"katex--inline\">3</span> .  In base  <span class=\"katex--inline\">9</span> , on the other hand, the same number is written as  <span class=\"katex--inline\">(2676)_{9}</span>  and ends in the digit  <span class=\"katex--inline\">6</span> .  For how many positive integers  <span class=\"katex--inline\">b</span>  does the base- <span class=\"katex--inline\">b</span> -representation of  <span class=\"katex--inline\">2013</span>  end in the digit  <span class=\"katex--inline\">3</span> ?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 6\\qquad\\textbf{(B)}\\ 9\\qquad\\textbf{(C)}\\ 13\\qquad\\textbf{(D)}\\ 16\\qquad\\textbf{(E)}\\ 18</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": "2013 AMC 10A Problem 19", "can_next": true, "can_prev": true, "nxt": "/problem/13_amc10A_p20", "prev": "/problem/13_amc10A_p18"}}