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project_euler/pe/p159.html
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project_euler/pe/p159.html
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<!DOCTYPE html>
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<html lang="en">
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<head>
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<meta charset="utf-8" />
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<meta name="author" content="Colin Hughes" />
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<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
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<meta name="keywords" content="programming,mathematics,problems,puzzles" />
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<title>Problem 159 - Project Euler</title>
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<div style="text-align:center;" class="print"><img src="images/pe_banner.png" alt="projecteuler.net" style="border:none;" /></div>
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<h2>Digital root sums of factorisations.
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</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 159</h3><span style="width:300px;color:#666;">Published on Saturday, 30th June 2007, 02:00 pm; Solved by 1635</span></div>
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<div class="problem_content" role="problem">
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<p>A composite number can be factored many different ways.
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For instance, not including multiplication by one, 24 can be factored in 7 distinct ways:</p>
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<div style="margin-left:50px;">
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24 = 2x2x2x3<br />
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24 = 2x3x4<br />
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24 = 2x2x6<br />
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24 = 4x6<br />
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24 = 3x8<br />
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24 = 2x12<br />
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24 = 24
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</div>
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<p>Recall that the digital root of a number, in base 10, is found by adding together the digits of that number,
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and repeating that process until a number is arrived at that is less than 10.
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Thus the digital root of 467 is 8.</p>
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<p>We shall call a Digital Root Sum (DRS) the sum of the digital roots of the individual factors of our number.<br />
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The chart below demonstrates all of the DRS values for 24.</p>
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<table align="center" border="1" cellpadding="2" cellspacing="0">
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<tr><th>Factorisation</th><th>Digital Root Sum</th></tr>
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<tr><td><div style="margin-left:20px;">2x2x2x3</div></td>
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<td><div style="text-align:center;">9</div></td></tr>
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<tr><td><div style="margin-left:20px;">2x3x4</div></td>
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<td><div style="text-align:center;">9</div></td></tr>
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<tr><td><div style="margin-left:20px;">2x2x6</div></td>
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<td><div style="text-align:center;">10</div></td></tr>
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<tr><td><div style="margin-left:20px;">4x6</div></td>
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<td><div style="text-align:center;">10</div></td></tr>
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<tr><td><div style="margin-left:20px;">3x8</div></td>
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<td><div style="text-align:center;">11</div></td></tr>
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<tr><td><div style="margin-left:20px;">2x12</div></td>
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<td><div style="text-align:center;">5</div></td></tr>
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<tr><td><div style="margin-left:20px;">24</div></td>
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<td><div style="text-align:center;">6</div></td></tr>
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</table>
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<p>The maximum Digital Root Sum of 24 is 11.<br />
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The function mdrs(<var>n</var>) gives the maximum Digital Root Sum of <var>n</var>. So mdrs(24)=11.<br />
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Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' />mdrs(<var>n</var>) for 1 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <var>n</var> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 1,000,000.</p>
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