{"status": "success", "data": {"description_md": "Consider the decimal expansion of a fraction $\\frac{1}{k}$, where $k > 1$ is a positive integer where $2 \\nmid k$ and $5 \\nmid k$. Now, it's decimal expansion will repeat after a certain number of digits, which we'll call it's period.\n\nFor example, $\\frac{1}{137}$ = $0.\\overline{00729927}$, so $\\frac{1}{137}$ has a period of $8$.\n\nFind the minimum value of $k$ such that $\\frac{1}{k}$ has a period of $8$.", "description_html": "<p>Consider the decimal expansion of a fraction <span class=\"katex--inline\">\\frac{1}{k}</span>, where <span class=\"katex--inline\">k &gt; 1</span> is a positive integer where <span class=\"katex--inline\">2 \\nmid k</span> and <span class=\"katex--inline\">5 \\nmid k</span>. Now, it&#8217;s decimal expansion will repeat after a certain number of digits, which we&#8217;ll call it&#8217;s period.</p>&#10;<p>For example, <span class=\"katex--inline\">\\frac{1}{137}</span> = <span class=\"katex--inline\">0.\\overline{00729927}</span>, so <span class=\"katex--inline\">\\frac{1}{137}</span> has a period of <span class=\"katex--inline\">8</span>.</p>&#10;<p>Find the minimum value of <span class=\"katex--inline\">k</span> such that <span class=\"katex--inline\">\\frac{1}{k}</span> has a period of <span class=\"katex--inline\">8</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Christmas Contest - Individual Round - Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/christmas1_individual-p13", "prev": "/problem/christmas1_individual-p11"}}