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<p>C++ Program to calculate the number of positive divisors.
<a href="#details">More...</a></p>
<div class="textblock"><code>#include &lt;cassert&gt;</code><br />
<code>#include &lt;iostream&gt;</code><br />
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Include dependency graph for number_of_positive_divisors.cpp:</div>
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<tr class="memitem:ad89ccced8504b5116046cfa03066ffeb"><td class="memItemLeft" align="right" valign="top">int&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d0/da2/number__of__positive__divisors_8cpp.html#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a> (int n)</td></tr>
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<tr class="memitem:a88ec9ad42717780d6caaff9d3d6977f9"><td class="memItemLeft" align="right" valign="top">void&#160;</td><td class="memItemRight" valign="bottom"><a class="el" href="../../d0/da2/number__of__positive__divisors_8cpp.html#a88ec9ad42717780d6caaff9d3d6977f9">tests</a> ()</td></tr>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
<div class="textblock"><p>C++ Program to calculate the number of positive divisors. </p>
<p>This algorithm uses the prime factorization approach. Any positive integer can be written as a product of its prime factors. <br />
Let \(N = p_1^{e_1} \times p_2^{e_2} \times\cdots\times p_k^{e_k}\) where \(p_1,\, p_2,\, \dots,\, p_k\) are distinct prime factors of \(N\) and \(e_1,\, e_2,\, \dots,\, e_k\) are respective positive integer exponents. <br />
Each positive divisor of \(N\) is in the form \(p_1^{g_1}\times p_2^{g_2}\times\cdots\times p_k^{g_k}\) where \(0\le g_i\le e_i\) are integers for all \(1\le i\le k\). <br />
Finally, there are \((e_1+1) \times (e_2+1)\times\cdots\times (e_k+1)\) positive divisors of \(N\) since we can choose every \(g_i\) independently.</p>
<p>Example: <br />
\(N = 36 = (3^2 \cdot 2^2)\) <br />
\(\mbox{number_of_positive_divisors}(36) = (2+1) \cdot (2+1) = 9\). <br />
list of positive divisors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36.</p>
<p>Similarly, for N = -36 the number of positive divisors remain same. </p>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">&#9670;&nbsp;</a></span>main()</h2>
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<p>Main function </p>
<div class="fragment"><div class="line"><a name="l00081"></a><span class="lineno"> 81</span>&#160; {</div>
<div class="line"><a name="l00082"></a><span class="lineno"> 82</span>&#160; <a class="code" href="../../d0/da2/number__of__positive__divisors_8cpp.html#a88ec9ad42717780d6caaff9d3d6977f9">tests</a>();</div>
<div class="line"><a name="l00083"></a><span class="lineno"> 83</span>&#160; <span class="keywordtype">int</span> n;</div>
<div class="line"><a name="l00084"></a><span class="lineno"> 84</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> &gt;&gt; n;</div>
<div class="line"><a name="l00085"></a><span class="lineno"> 85</span>&#160; <span class="keywordflow">if</span> (n == 0) {</div>
<div class="line"><a name="l00086"></a><span class="lineno"> 86</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;All non-zero numbers are divisors of 0 !&quot;</span> &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00087"></a><span class="lineno"> 87</span>&#160; } <span class="keywordflow">else</span> {</div>
<div class="line"><a name="l00088"></a><span class="lineno"> 88</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <span class="stringliteral">&quot;Number of positive divisors is : &quot;</span>;</div>
<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>&#160; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> &lt;&lt; <a class="code" href="../../d0/da2/number__of__positive__divisors_8cpp.html#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(n) &lt;&lt; <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>&#160; }</div>
<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>&#160; <span class="keywordflow">return</span> 0;</div>
<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>&#160;}</div>
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<h2 class="memtitle"><span class="permalink"><a href="#ad89ccced8504b5116046cfa03066ffeb">&#9670;&nbsp;</a></span>number_of_positive_divisors()</h2>
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<td class="memname">int number_of_positive_divisors </td>
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<p>Function to compute the number of positive divisors. </p><dl class="params"><dt>Parameters</dt><dd>
<table class="params">
<tr><td class="paramname">n</td><td>number to compute divisors for </td></tr>
</table>
</dd>
</dl>
<dl class="section return"><dt>Returns</dt><dd>number of positive divisors of n (or 1 if n = 0) </dd></dl>
<div class="fragment"><div class="line"><a name="l00033"></a><span class="lineno"> 33</span>&#160; {</div>
<div class="line"><a name="l00034"></a><span class="lineno"> 34</span>&#160; <span class="keywordflow">if</span> (n &lt; 0) {</div>
<div class="line"><a name="l00035"></a><span class="lineno"> 35</span>&#160; n = -n; <span class="comment">// take the absolute value of n</span></div>
<div class="line"><a name="l00036"></a><span class="lineno"> 36</span>&#160; }</div>
<div class="line"><a name="l00037"></a><span class="lineno"> 37</span>&#160; </div>
<div class="line"><a name="l00038"></a><span class="lineno"> 38</span>&#160; <span class="keywordtype">int</span> number_of_divisors = 1;</div>
<div class="line"><a name="l00039"></a><span class="lineno"> 39</span>&#160; </div>
<div class="line"><a name="l00040"></a><span class="lineno"> 40</span>&#160; <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 2; i * i &lt;= n; i++) {</div>
<div class="line"><a name="l00041"></a><span class="lineno"> 41</span>&#160; <span class="comment">// This part is doing the prime factorization.</span></div>
<div class="line"><a name="l00042"></a><span class="lineno"> 42</span>&#160; <span class="comment">// Note that we cannot find a composite divisor of n unless we would</span></div>
<div class="line"><a name="l00043"></a><span class="lineno"> 43</span>&#160; <span class="comment">// already previously find the corresponding prime divisor and dvided</span></div>
<div class="line"><a name="l00044"></a><span class="lineno"> 44</span>&#160; <span class="comment">// n by that prime. Therefore, all the divisors found here will</span></div>
<div class="line"><a name="l00045"></a><span class="lineno"> 45</span>&#160; <span class="comment">// actually be primes.</span></div>
<div class="line"><a name="l00046"></a><span class="lineno"> 46</span>&#160; <span class="comment">// The loop terminates early when it is left with a number n which</span></div>
<div class="line"><a name="l00047"></a><span class="lineno"> 47</span>&#160; <span class="comment">// does not have a divisor smaller or equal to sqrt(n) - that means</span></div>
<div class="line"><a name="l00048"></a><span class="lineno"> 48</span>&#160; <span class="comment">// the remaining number is a prime itself.</span></div>
<div class="line"><a name="l00049"></a><span class="lineno"> 49</span>&#160; <span class="keywordtype">int</span> prime_exponent = 0;</div>
<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>&#160; <span class="keywordflow">while</span> (n % i == 0) {</div>
<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>&#160; <span class="comment">// Repeatedly divide n by the prime divisor n to compute</span></div>
<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>&#160; <span class="comment">// the exponent (e_i in the algorithm description).</span></div>
<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>&#160; prime_exponent++;</div>
<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>&#160; n /= i;</div>
<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>&#160; }</div>
<div class="line"><a name="l00056"></a><span class="lineno"> 56</span>&#160; number_of_divisors *= prime_exponent + 1;</div>
<div class="line"><a name="l00057"></a><span class="lineno"> 57</span>&#160; }</div>
<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>&#160; <span class="keywordflow">if</span> (n &gt; 1) {</div>
<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>&#160; <span class="comment">// In case the remaining number n is a prime number itself</span></div>
<div class="line"><a name="l00060"></a><span class="lineno"> 60</span>&#160; <span class="comment">// (essentially p_k^1) the final answer is also multiplied by (e_k+1).</span></div>
<div class="line"><a name="l00061"></a><span class="lineno"> 61</span>&#160; number_of_divisors *= 2;</div>
<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>&#160; }</div>
<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>&#160; </div>
<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>&#160; <span class="keywordflow">return</span> number_of_divisors;</div>
<div class="line"><a name="l00065"></a><span class="lineno"> 65</span>&#160;}</div>
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<p>Test implementations </p>
<div class="fragment"><div class="line"><a name="l00070"></a><span class="lineno"> 70</span>&#160; {</div>
<div class="line"><a name="l00071"></a><span class="lineno"> 71</span>&#160; assert(<a class="code" href="../../d0/da2/number__of__positive__divisors_8cpp.html#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(36) == 9);</div>
<div class="line"><a name="l00072"></a><span class="lineno"> 72</span>&#160; assert(<a class="code" href="../../d0/da2/number__of__positive__divisors_8cpp.html#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(-36) == 9);</div>
<div class="line"><a name="l00073"></a><span class="lineno"> 73</span>&#160; assert(<a class="code" href="../../d0/da2/number__of__positive__divisors_8cpp.html#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(1) == 1);</div>
<div class="line"><a name="l00074"></a><span class="lineno"> 74</span>&#160; assert(<a class="code" href="../../d0/da2/number__of__positive__divisors_8cpp.html#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(2011) == 2); <span class="comment">// 2011 is a prime</span></div>
<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>&#160; assert(<a class="code" href="../../d0/da2/number__of__positive__divisors_8cpp.html#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(756) == 24); <span class="comment">// 756 = 2^2 * 3^3 * 7</span></div>
<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>&#160;}</div>
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