{"status": "success", "data": {"description_md": "Define the *number complement* of a positive integer $n$ in base $b$ as a new number $(n^\\prime)_b$, where the sum of the corresponding digits of $n$ and $n^\\prime$ is equal to $b-1$. For example, the number complement of $(126)_9$ is $(762)_9$ because $1+7 = 2+6 = 6+2 = 8$. Find the number of base $8$ positive integers $k$ with a digit sum of $32$ and each digit $d \\in \\{1, 2, 3, 4, 5, 6\\}$, such the digit sum of $k$'s number complement is equal to $10$.  \n  \n$\\textbf{(A)}~252\\qquad\\textbf{(B)}~210\\qquad\\textbf{(C)}~84\\qquad\\textbf{(D)}~126\\qquad\\textbf{(E)}~462$", "description_html": "<p>Define the <em>number complement</em> of a positive integer <span class=\"katex--inline\">n</span> in base <span class=\"katex--inline\">b</span> as a new number <span class=\"katex--inline\">(n^\\prime)_b</span>, where the sum of the corresponding digits of <span class=\"katex--inline\">n</span> and <span class=\"katex--inline\">n^\\prime</span> is equal to <span class=\"katex--inline\">b-1</span>. For example, the number complement of <span class=\"katex--inline\">(126)_9</span> is <span class=\"katex--inline\">(762)_9</span> because <span class=\"katex--inline\">1+7 = 2+6 = 6+2 = 8</span>. Find the number of base <span class=\"katex--inline\">8</span> positive integers <span class=\"katex--inline\">k</span> with a digit sum of <span class=\"katex--inline\">32</span> and each digit <span class=\"katex--inline\">d \\in \\{1, 2, 3, 4, 5, 6\\}</span>, such the digit sum of <span class=\"katex--inline\">k</span>'s number complement is equal to <span class=\"katex--inline\">10</span>.</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~252\\qquad\\textbf{(B)}~210\\qquad\\textbf{(C)}~84\\qquad\\textbf{(D)}~126\\qquad\\textbf{(E)}~462</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "2024 Mock AMC 10 - Problem 18", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}