99 lines
6.4 KiB
HTML
99 lines
6.4 KiB
HTML
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<h2>Cyclic numbers</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 358</h3><span style="width:300px;color:#666;">Published on Saturday, 12th November 2011, 07:00 pm; Solved by 704</span></div>
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<p>A <b>cyclic number</b> with <var>n</var> digits has a very interesting property:<br/>
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When it is multiplied by 1, 2, 3, 4, ... <var>n</var>, all the products have exactly the same digits, in the same order, but rotated in a circular fashion!
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<p>
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The smallest cyclic number is the 6-digit number 142857 :<br/>
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142857 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 1 = 142857<br/>
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142857 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 2 = 285714<br/>
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142857 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 3 = 428571<br/>
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142857 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 4 = 571428<br/>
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142857 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 5 = 714285<br/>
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142857 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 6 = 857142
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</p>
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<p>
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The next cyclic number is 0588235294117647 with 16 digits :<br/>
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0588235294117647 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 1 = 0588235294117647<br/>
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0588235294117647 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 2 = 1176470588235294<br/>
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0588235294117647 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 3 = 1764705882352941<br/>
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...<br/>
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0588235294117647 <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> 16 = 9411764705882352
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<p>
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Note that for cyclic numbers, leading zeros are important.
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<p>
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There is only one cyclic number for which, the eleven leftmost digits are 00000000137 and the five rightmost digits are 56789 (i.e., it has the form 00000000137...56789 with an unknown number of digits in the middle). Find the sum of all its digits.
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