139 lines
9.3 KiB
HTML
139 lines
9.3 KiB
HTML
<!DOCTYPE html>
|
|
<html lang="en">
|
|
<head>
|
|
<meta charset="utf-8" />
|
|
<meta name="author" content="Colin Hughes" />
|
|
<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
|
|
<meta name="keywords" content="programming,mathematics,problems,puzzles" />
|
|
|
|
<title>Problem 124 - Project Euler</title>
|
|
<link rel="shortcut icon" href="http://projecteuler.net/favicon.ico" />
|
|
<link rel="stylesheet" type="text/css" href="style_main.css" />
|
|
<link rel="stylesheet" type="text/css" href="style_light.css" />
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
jax: ["input/TeX", "output/HTML-CSS"],
|
|
tex2jax: {
|
|
inlineMath: [ ["$","$"], ["\\(","\\)"] ],
|
|
displayMath: [ ["$$","$$"], ["\\[","\\]"] ],
|
|
processEscapes: true
|
|
},
|
|
"HTML-CSS": { availableFonts: ["TeX"] }
|
|
});
|
|
</script>
|
|
|
|
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML">
|
|
</script>
|
|
</head>
|
|
|
|
<body>
|
|
<div id="container">
|
|
|
|
<div id="nav" class="noprint">
|
|
<ul>
|
|
<li><a href="about" title="About" accesskey="h">About</a></li>
|
|
<li><a href="register" title="Register" accesskey="1">Register</a></li>
|
|
<li id="current"><a href="problems" title="Problems" accesskey="2">Problems</a></li>
|
|
<li><a href="login" title="Login" accesskey="3">Login</a></li>
|
|
</ul>
|
|
</div>
|
|
<div id="info_panel"><a href="rss2_euler.xml"><img src="images/icon_rss.png" alt="RSS Feed" title="RSS Feed" /></a><a href="secure=ef99f"><img src="images/icon_lock.png" alt="Use secure connection" title="Use secure connection" /></a></div>
|
|
|
|
<div id="logo" class="noprint">
|
|
<img src="images/pe_banner_light.png" alt="Project Euler .net" />
|
|
</div>
|
|
|
|
<div id="content">
|
|
<div style="text-align:center;" class="print"><img src="images/pe_banner.png" alt="projecteuler.net" style="border:none;" /></div>
|
|
<h2>Ordered radicals</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 124</h3><span style="width:300px;color:#666;">Published on Friday, 14th July 2006, 06:00 pm; Solved by 7220</span></div>
|
|
<div class="problem_content" role="problem">
|
|
|
|
<p>The radical of <i>n</i>, rad(<i>n</i>), is the product of distinct prime factors of <i>n</i>. For example, 504 = 2<sup>3</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 3<sup>2</sup> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 7, so rad(504) = 2 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 3 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 7 = 42.</p>
|
|
<p>If we calculate rad(<i>n</i>) for <i>1</i> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <i>n</i> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 10, then sort them on rad(<i>n</i>), and sorting on <i>n</i> if the radical values are equal, we get:</p>
|
|
<table cellpadding='2' cellspacing='0' border='0' align='center'>
|
|
<tr>
|
|
<td colspan='2'><div style='text-align:center;'><b>Unsorted</b></div></td>
|
|
<td> </td>
|
|
<td colspan='3'><div style='text-align:center;'><b>Sorted</b></div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'><img src='images/spacer.gif' width='50' height='1' alt='' /><br /><b><i>n</i></b></div></td>
|
|
<td><div style='text-align:center;'><img src='images/spacer.gif' width='50' height='1' alt='' /><br /><b>rad(<i>n</i>)</b></div></td>
|
|
<td><img src='images/spacer.gif' width='50' height='1' alt='' /><br /></td>
|
|
<td><div style='text-align:center;'><img src='images/spacer.gif' width='50' height='1' alt='' /><br /><b><i>n</i></b></div></td>
|
|
<td><div style='text-align:center;'><img src='images/spacer.gif' width='50' height='1' alt='' /><br /><b>rad(<i>n</i>)</b></div></td>
|
|
<td><div style='text-align:center;'><img src='images/spacer.gif' width='50' height='1' alt='' /><br /><b>k</b></div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>1</div></td><td><div style='text-align:center;'>1</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>1</div></td><td><div style='text-align:center;'>1</div></td><td><div style='text-align:center;'>1</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>2</div></td><td><div style='text-align:center;'>2</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>2</div></td><td><div style='text-align:center;'>2</div></td><td><div style='text-align:center;'>2</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>3</div></td><td><div style='text-align:center;'>3</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>4</div></td><td><div style='text-align:center;'>2</div></td><td><div style='text-align:center;'>3</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>4</div></td><td><div style='text-align:center;'>2</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>8</div></td><td><div style='text-align:center;'>2</div></td><td><div style='text-align:center;'>4</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>5</div></td><td><div style='text-align:center;'>5</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>3</div></td><td><div style='text-align:center;'>3</div></td><td><div style='text-align:center;'>5</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>6</div></td><td><div style='text-align:center;'>6</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>9</div></td><td><div style='text-align:center;'>3</div></td><td><div style='text-align:center;'>6</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>7</div></td><td><div style='text-align:center;'>7</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>5</div></td><td><div style='text-align:center;'>5</div></td><td><div style='text-align:center;'>7</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>8</div></td><td><div style='text-align:center;'>2</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>6</div></td><td><div style='text-align:center;'>6</div></td><td><div style='text-align:center;'>8</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>9</div></td><td><div style='text-align:center;'>3</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>7</div></td><td><div style='text-align:center;'>7</div></td><td><div style='text-align:center;'>9</div></td>
|
|
</tr>
|
|
<tr>
|
|
<td><div style='text-align:center;'>10</div></td><td><div style='text-align:center;'>10</div></td>
|
|
<td> </td>
|
|
<td><div style='text-align:center;'>10</div></td><td><div style='text-align:center;'>10</div></td><td><div style='text-align:center;'>10</div></td>
|
|
</tr>
|
|
</table>
|
|
<p>Let E(<i>k</i>) be the <i>k</i>th element in the sorted <i>n</i> column; for example, E(4) = 8 and E(6) = 9.</p>
|
|
<p>If rad(<i>n</i>) is sorted for 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> <i>n</i> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 100000, find E(10000).</p>
|
|
|
|
</div><br />
|
|
<br /></div>
|
|
|
|
|
|
|
|
|
|
|
|
<div id="footer" class="noprint">
|
|
<a href="copyright">Project Euler Copyright Information</a>
|
|
<!--/Creative Commons License--><!-- <rdf:RDF xmlns="http://web.resource.org/cc/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#">
|
|
<Work rdf:about="">
|
|
<license rdf:resource="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/" />
|
|
<dc:type rdf:resource="http://purl.org/dc/dcmitype/Text" />
|
|
</Work>
|
|
<License rdf:about="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/"><permits rdf:resource="http://web.resource.org/cc/Reproduction"/><permits rdf:resource="http://web.resource.org/cc/Distribution"/><requires rdf:resource="http://web.resource.org/cc/Notice"/><requires rdf:resource="http://web.resource.org/cc/Attribution"/><prohibits rdf:resource="http://web.resource.org/cc/CommercialUse"/><permits rdf:resource="http://web.resource.org/cc/DerivativeWorks"/><requires rdf:resource="http://web.resource.org/cc/ShareAlike"/></License></rdf:RDF> -->
|
|
</div>
|
|
</div>
|
|
<div style="height:1px;"> </div></body>
|
|
</html> |