137 lines
5.7 KiB
HTML
137 lines
5.7 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 69 - 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=29482"><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>Totient maximum</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 69</h3><span style="width:300px;color:#666;">Published on Friday, 7th May 2004, 06:00 pm; Solved by 16210</span></div>
|
|
<div class="problem_content" role="problem">
|
|
|
|
<p>Euler's Totient function, φ(<i>n</i>) [sometimes called the phi function], is used to determine the number of numbers less than <i>n</i> which are relatively prime to <i>n</i>. For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6.</p>
|
|
<div style='margin-left:100px;'>
|
|
<table border='1'>
|
|
<tr>
|
|
<td><b><i>n</i></b></td>
|
|
<td><b>Relatively Prime</b></td>
|
|
<td><b>φ(<i>n</i>)</b></td>
|
|
<td><b><i>n</i>/φ(<i>n</i>)</b></td>
|
|
</tr>
|
|
<tr>
|
|
<td>2</td>
|
|
<td>1</td>
|
|
<td>1</td>
|
|
<td>2</td>
|
|
</tr>
|
|
<tr>
|
|
<td>3</td>
|
|
<td>1,2</td>
|
|
<td>2</td>
|
|
<td>1.5</td>
|
|
</tr>
|
|
<tr>
|
|
<td>4</td>
|
|
<td>1,3</td>
|
|
<td>2</td>
|
|
<td>2</td>
|
|
</tr>
|
|
<tr>
|
|
<td>5</td>
|
|
<td>1,2,3,4</td>
|
|
<td>4</td>
|
|
<td>1.25</td>
|
|
</tr>
|
|
<tr>
|
|
<td>6</td>
|
|
<td>1,5</td>
|
|
<td>2</td>
|
|
<td>3</td>
|
|
</tr>
|
|
<tr>
|
|
<td>7</td>
|
|
<td>1,2,3,4,5,6</td>
|
|
<td>6</td>
|
|
<td>1.1666...</td>
|
|
</tr>
|
|
<tr>
|
|
<td>8</td>
|
|
<td>1,3,5,7</td>
|
|
<td>4</td>
|
|
<td>2</td>
|
|
</tr>
|
|
<tr>
|
|
<td>9</td>
|
|
<td>1,2,4,5,7,8</td>
|
|
<td>6</td>
|
|
<td>1.5</td>
|
|
</tr>
|
|
<tr>
|
|
<td>10</td>
|
|
<td>1,3,7,9</td>
|
|
<td>4</td>
|
|
<td>2.5</td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
<p>It can be seen that <i>n</i>=6 produces a maximum <i>n</i>/φ(<i>n</i>) for <i>n</i> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 10.</p>
|
|
<p>Find the value of <i>n</i> <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 1,000,000 for which <i>n</i>/φ(<i>n</i>) is a maximum.</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> |