{"status": "success", "data": {"description_md": "Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\\overline{abcd},$ where at least one of the digits $a, b, c,$ or $d$ is nonzero. Let $N$ be the number of distinct numerators when numbers in $S$ are written as fractions in lowest terms. For example, both $4$ and $410$ are counted among the distinct numerators for numbers in $S$ because $0.\\overline{3636} = \\frac{4}{11}$ and $0.\\overline{1230} = \\frac{410}{3333}.$ Find the remainder when $N$ is divided by $1000.$\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full 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\">S</span> be the set of all rational numbers that can be expressed as a repeating decimal in the form <span class=\"katex--inline\">0.\\overline{abcd},</span> where at least one of the digits <span class=\"katex--inline\">a, b, c,</span> or <span class=\"katex--inline\">d</span> is nonzero. Let <span class=\"katex--inline\">N</span> be the number of distinct numerators when numbers in <span class=\"katex--inline\">S</span> are written as fractions in lowest terms. For example, both <span class=\"katex--inline\">4</span> and <span class=\"katex--inline\">410</span> are counted among the distinct numerators for numbers in <span class=\"katex--inline\">S</span> because <span class=\"katex--inline\">0.\\overline{3636} = \\frac{4}{11}</span> and <span class=\"katex--inline\">0.\\overline{1230} = \\frac{410}{3333}.</span> Find the remainder when <span class=\"katex--inline\">N</span> is divided by <span class=\"katex--inline\">1000.</span></p>&#10;<hr/>&#10;<p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. 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": 6, "problem_name": "2022 AIME I Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/22_aime_I_p14", "prev": "/problem/22_aime_I_p12"}}