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puzzles/project_euler/pe/p183.html
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puzzles/project_euler/pe/p183.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="keywords" content="programming,mathematics,problems,puzzles" />
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<title>Problem 183 - Project Euler</title>
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<h2>Maximum product of parts</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 183</h3><span style="width:300px;color:#666;">Published on Friday, 22nd February 2008, 05:00 pm; Solved by 2612</span></div>
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<div class="problem_content" role="problem">
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<p>Let N be a positive integer and let N be split into <var>k</var> equal parts, <var>r</var> = N/<var>k</var>, so that N = <var>r</var> + <var>r</var> + ... + <var>r</var>.<br />
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Let P be the product of these parts, P = <var>r</var> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <var>r</var> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> ... <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> <var>r</var> = <var>r</var><sup><var>k</var></sup>.</p>
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<p>For example, if 11 is split into five equal parts, 11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2, then P = 2.2<sup>5</sup> = 51.53632.</p>
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<p>Let M(N) = P<sub>max</sub> for a given value of N.</p>
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<p>It turns out that the maximum for N = 11 is found by splitting eleven into four equal parts which leads to P<sub>max</sub> = (11/4)<sup>4</sup>; that is, M(11) = 14641/256 = 57.19140625, which is a terminating decimal.</p>
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<p>However, for N = 8 the maximum is achieved by splitting it into three equal parts, so M(8) = 512/27, which is a non-terminating decimal.</p>
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<p>Let D(N) = N if M(N) is a non-terminating decimal and D(N) = -N if M(N) is a terminating decimal.</p>
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<p>For example, ΣD(N) for 5 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> N <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 100 is 2438.</p>
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<p>Find ΣD(N) for 5 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> N <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> 10000.</p>
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