86 lines
6.4 KiB
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86 lines
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<h2>abc-hits</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 127</h3><span style="width:300px;color:#666;">Published on Friday, 1st September 2006, 06:00 pm; Solved by 2788</span></div>
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<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>
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<p>We shall define the triplet of positive integers (<i>a</i>, <i>b</i>, <i>c</i>) to be an abc-hit if:</p>
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<ol>
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<li>GCD(<i>a,</i> <i>b</i>) = GCD(<i>a</i>, <i>c</i>) = GCD(<i>b</i>, <i>c</i>) = 1</li>
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<li><i>a</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <i>b</i></li>
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<li><i>a</i> + <i>b</i> = <i>c</i></li>
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<li>rad(<i>abc</i>) <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> <i>c</i></li>
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</ol>
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<p>For example, (5, 27, 32) is an abc-hit, because:</p>
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<ol>
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<li>GCD(5, 27) = GCD(5, 32) = GCD(27, 32) = 1</li>
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<li>5 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 27</li>
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<li>5 + 27 = 32</li>
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<li>rad(4320) = 30 <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 32</li>
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<p>It turns out that abc-hits are quite rare and there are only thirty-one abc-hits for <i>c</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 1000, with <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /><i>c</i> = 12523.</p>
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<p>Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /><i>c</i> for <i>c</i> <img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /> 120000.</p>
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