74 lines
6.1 KiB
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74 lines
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<h2>Balanced Numbers</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 217</h3><span style="width:300px;color:#666;">Published on Friday, 14th November 2008, 09:00 pm; Solved by 831</span></div>
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A positive integer with <var>k</var> (decimal) digits is called balanced if its first <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' /><sup><var>k</var></sup>/<sub>2</sub><img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' /> digits sum to the same value as its last <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' /><sup><var>k</var></sup>/<sub>2</sub><img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' /> digits, where <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' /><var>x</var><img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' />, pronounced <span style="font-style: italic">ceiling</span> of <var>x</var>, is the smallest integer <img src='images/symbol_ge.gif' width='10' height='12' alt='≥' border='0' style='vertical-align:middle;' /> <var>x</var>, thus <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' />π<img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' /> = 4 and <img src='images/symbol_lceil.gif' width='6' height='16' alt='⌈' border='0' style='vertical-align:middle;' />5<img src='images/symbol_rceil.gif' width='6' height='16' alt='⌉' border='0' style='vertical-align:middle;' /> = 5.</p>
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<p>So, for example, all palindromes are balanced, as is 13722.</p>
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<p>Let T(<var>n</var>) be the sum of all balanced numbers less than 10<sup><var>n</var></sup>. <br />
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Thus: T(1) = 45, T(2) = 540 and T(5) = 334795890. </p>
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<p>Find T(47) mod 3<sup>15</sup></p>
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