PAT/source/html/t1018.html

Subnumbers
<div class="ques-view"><p>Given a positive integer <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>, let us define a <strong>subnumber</strong> of <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> as a consecutive number of digits <strong>NOT starting with 0</strong>. For example if <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi><mo>=</mo><mn>1</mn><mn>0</mn><mn>2</mn><mn>1</mn></mrow>N = 1021</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mrel">=</span><span class="mord mathrm">1</span><span class="mord mathrm">0</span><span class="mord mathrm">2</span><span class="mord mathrm">1</span></span></span></span>, it has 7 subnumbers, namely, 1, 10, 102, 1021, 2, 21 and 1 (again). Here is your task: please calculate the sum of all the subnumbers of <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>. For 1021, the sum is 1+10+102+1021+2+21+1 = 1158. Since the result may be very large, output the answer modulo 1000000007 (<span class="katex"><span class="katex-mathml"><math><mrow><mn>1</mn><msup><mn>0</mn><mn>9</mn></msup><mo>+</mo><mn>7</mn></mrow>10^9 + 7</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.897438em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mord"><span class="mord mathrm">0</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathrm mtight">9</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">+</span><span class="mord mathrm">7</span></span></span></span>) please.</p>
<h3 id="input-specification-">Input Specification:</h3>
<p>Each input file contains one test case, which gives the integer <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><mrow><mn>0</mn><mo>&lt;</mo><mi>N</mi><mo>&lt;</mo><mn>1</mn><msup><mn>0</mn><mrow><mn>1</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn><mn>0</mn></mrow></msup></mrow>0 &lt; N &lt; 10^{100000}</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.853208em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">0</span><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.10903em;">N</span><span class="mrel">&lt;</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathrm">0</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord scriptstyle uncramped mtight"><span class="mord mathrm mtight">1</span><span class="mord mathrm mtight">0</span><span class="mord mathrm mtight">0</span><span class="mord mathrm mtight">0</span><span class="mord mathrm mtight">0</span><span class="mord mathrm mtight">0</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>) in a line.</p>
<h3 id="output-specification-">Output Specification:</h3>
<p>Print in a line the sum of all <span class="katex"><span class="katex-mathml"><math><mrow><mi>N</mi></mrow>N</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.68333em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.10903em;">N</span></span></span></span>'s subnumbers (modulo 1000000007).</p>
<h3 id="sample-input-">Sample Input:</h3>
<pre><code class="lang-in">1234567890123456789
</code></pre>
<h3 id="sample-output-">Sample Output:</h3>
<pre><code class="lang-out">332876913
</code></pre>
</div>