75 lines
5.5 KiB
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75 lines
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<h2>Composites with prime repunit property</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 130</h3><span style="width:300px;color:#666;">Published on Friday, 27th October 2006, 06:00 pm; Solved by 2849</span></div>
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<p>A number consisting entirely of ones is called a repunit. We shall define R(<i>k</i>) to be a repunit of length <i>k</i>; for example, R(6) = 111111.</p>
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<p>Given that <i>n</i> is a positive integer and GCD(<i>n</i>, 10) = 1, it can be shown that there always exists a value, <i>k</i>, for which R(<i>k</i>) is divisible by <i>n</i>, and let A(<i>n</i>) be the least such value of <i>k</i>; for example, A(7) = 6 and A(41) = 5.</p>
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<p>You are given that for all primes, <i>p</i> <img src='images/symbol_gt.gif' width='10' height='10' alt='>' border='0' style='vertical-align:middle;' /> 5, that <i>p</i> <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 1 is divisible by A(<i>p</i>). For example, when <i>p</i> = 41, A(41) = 5, and 40 is divisible by 5.</p>
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<p>However, there are rare composite values for which this is also true; the first five examples being 91, 259, 451, 481, and 703.</p>
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<p>Find the sum of the first twenty-five composite values of <i>n</i> for which<br />GCD(<i>n</i>, 10) = 1 and <i>n</i> <img src='images/symbol_minus.gif' width='9' height='3' alt='−' border='0' style='vertical-align:middle;' /> 1 is divisible by A(<i>n</i>).</p>
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