{"status": "success", "data": {"description_md": "A positive integer has $k$ *trailing zeros* if its last $k$ digits are all zero and it has a non-zero digit immediately to the left of these $k$ zeros. For example, the number $1\\,030\\,000$ has 4 trailing zeros. Define $Z(m)$ to be the number of trailing zeros of the positive integer $m$. Lloyd is bored one day, so makes a list of the value of $n - Z(n!)$ for each integer $n$ from 100 to 10\\,000, inclusive. How many integers appear in his list at least three times?\n\n(Note: If $n$ is a positive integer, the symbol $n!$ (read \"$n$ factorial\") is used to represent the product of the integers from 1 to $n$. That is, $n! = n(n - 1)(n - 2)\\cdots(3)(2)(1)$. For example, $5! = 5(4)(3)(2)(1)$ or $5! = 120$.)\n\n$\\text{(A)}\\ 2\\qquad\\text{(B)}\\ 3\\qquad\\text{(C)}\\ 4\\qquad\\text{(D)}\\ 5\\qquad\\text{(E)}\\ 6$\n\n___\n\n**[Cayley](https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf), by [CEMC](https://cemc.uwaterloo.ca/), is licensed under the [CC BY-NC 3.0 DEED](https://creativecommons.org/licenses/by-nc/3.0/). They are not affiliated with or endorse TopsOJ in any way.**\n", "description_html": "<p>A positive integer has <span class=\"katex--inline\">k</span> *trailing zeros* if its last <span class=\"katex--inline\">k</span> digits are all zero and it has a non-zero digit immediately to the left of these <span class=\"katex--inline\">k</span> zeros. For example, the number <span class=\"katex--inline\">1\\,030\\,000</span> has 4 trailing zeros. Define <span class=\"katex--inline\">Z(m)</span> to be the number of trailing zeros of the positive integer <span class=\"katex--inline\">m</span>. Lloyd is bored one day, so makes a list of the value of <span class=\"katex--inline\">n - Z(n!)</span> for each integer <span class=\"katex--inline\">n</span> from 100 to 10\\,000, inclusive. How many integers appear in his list at least three times?</p>\n<p>(Note: If <span class=\"katex--inline\">n</span> is a positive integer, the symbol <span class=\"katex--inline\">n!</span> (read \"<span class=\"katex--inline\">n</span> factorial\") is used to represent the product of the integers from 1 to <span class=\"katex--inline\">n</span>. That is, <span class=\"katex--inline\">n! = n(n - 1)(n - 2)\\cdots(3)(2)(1)</span>. For example, <span class=\"katex--inline\">5! = 5(4)(3)(2)(1)</span> or <span class=\"katex--inline\">5! = 120</span>.)</p>\n<p><span class=\"katex--inline\">\\text{(A)}\\ 2\\qquad\\text{(B)}\\ 3\\qquad\\text{(C)}\\ 4\\qquad\\text{(D)}\\ 5\\qquad\\text{(E)}\\ 6</span></p>\n<hr>\n<p><strong><a href=\"https://www.cemc.uwaterloo.ca/contests/past_contests.html#pcf\">Cayley</a>, by <a href=\"https://cemc.uwaterloo.ca/\">CEMC</a>, is licensed under the <a href=\"https://creativecommons.org/licenses/by-nc/3.0/\">CC BY-NC 3.0 DEED</a>. They are not affiliated with or endorse TopsOJ in any way.</strong></p>\n", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2013 Cayley Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/13_cayley_p24"}}