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<a href="#func-members">Functions</a> </div>
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<div class="headertitle"><div class="title">number_of_positive_divisors.cpp File Reference</div></div>
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<p>C++ Program to calculate the number of positive divisors.
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<a href="#details">More...</a></p>
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<div class="textblock"><code>#include <cassert></code><br />
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<code>#include <iostream></code><br />
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</div><div class="textblock"><div class="dynheader">
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Include dependency graph for number_of_positive_divisors.cpp:</div>
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<div class="center"><iframe scrolling="no" frameborder="0" src="../../d4/da4/number__of__positive__divisors_8cpp__incl.svg" width="179" height="126"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe></div>
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Functions</h2></td></tr>
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<tr class="memitem:ad89ccced8504b5116046cfa03066ffeb" id="r_ad89ccced8504b5116046cfa03066ffeb"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a> (int n)</td></tr>
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<tr class="separator:ad89ccced8504b5116046cfa03066ffeb"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:a88ec9ad42717780d6caaff9d3d6977f9" id="r_a88ec9ad42717780d6caaff9d3d6977f9"><td class="memItemLeft" align="right" valign="top">void </td><td class="memItemRight" valign="bottom"><a class="el" href="#a88ec9ad42717780d6caaff9d3d6977f9">tests</a> ()</td></tr>
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<tr class="separator:a88ec9ad42717780d6caaff9d3d6977f9"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:ae66f6b31b5ad750f1fe042a706a4e3d4" id="r_ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="#ae66f6b31b5ad750f1fe042a706a4e3d4">main</a> ()</td></tr>
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<tr class="separator:ae66f6b31b5ad750f1fe042a706a4e3d4"><td class="memSeparator" colspan="2"> </td></tr>
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</table>
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<a name="details" id="details"></a><h2 class="groupheader">Detailed Description</h2>
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<div class="textblock"><p>C++ Program to calculate the number of positive divisors. </p>
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<p>This algorithm uses the prime factorization approach. Any positive integer can be written as a product of its prime factors. <br />
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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 />
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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 />
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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>
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<p>Example: <br />
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\(N = 36 = (3^2 \cdot 2^2)\) <br />
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\(\mbox{number_of_positive_divisors}(36) = (2+1) \cdot (2+1) = 9\). <br />
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list of positive divisors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36.</p>
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<p>Similarly, for N = -36 the number of positive divisors remain same. </p>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="ae66f6b31b5ad750f1fe042a706a4e3d4" name="ae66f6b31b5ad750f1fe042a706a4e3d4"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ae66f6b31b5ad750f1fe042a706a4e3d4">◆ </a></span>main()</h2>
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<div class="memitem">
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<div class="memproto">
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<td class="memname">int main </td>
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<td>(</td>
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<td class="paramtype">void</td> <td class="paramname"><span class="paramname"><em></em></span></td><td>)</td>
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<td></td>
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</div><div class="memdoc">
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<p>Main function </p>
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<div class="fragment"><div class="line"><span class="lineno"> 81</span> {</div>
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<div class="line"><span class="lineno"> 82</span> <a class="code hl_function" href="#a88ec9ad42717780d6caaff9d3d6977f9">tests</a>();</div>
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<div class="line"><span class="lineno"> 83</span> <span class="keywordtype">int</span> n;</div>
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<div class="line"><span class="lineno"> 84</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a> >> n;</div>
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<div class="line"><span class="lineno"> 85</span> <span class="keywordflow">if</span> (n == 0) {</div>
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<div class="line"><span class="lineno"> 86</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"All non-zero numbers are divisors of 0 !"</span> << <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><span class="lineno"> 87</span> } <span class="keywordflow">else</span> {</div>
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<div class="line"><span class="lineno"> 88</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Number of positive divisors is : "</span>;</div>
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<div class="line"><span class="lineno"> 89</span> <a class="code hl_classRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <a class="code hl_function" href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(n) << <a class="code hl_functionRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><span class="lineno"> 90</span> }</div>
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<div class="line"><span class="lineno"> 91</span> <span class="keywordflow">return</span> 0;</div>
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<div class="line"><span class="lineno"> 92</span>}</div>
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<div class="ttc" id="abasic_istream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_istream.html">std::cin</a></div></div>
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<div class="ttc" id="abasic_ostream_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a></div></div>
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<div class="ttc" id="aendl_html"><div class="ttname"><a href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a></div><div class="ttdeci">T endl(T... args)</div></div>
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<div class="ttc" id="anumber__of__positive__divisors_8cpp_html_a88ec9ad42717780d6caaff9d3d6977f9"><div class="ttname"><a href="#a88ec9ad42717780d6caaff9d3d6977f9">tests</a></div><div class="ttdeci">void tests()</div><div class="ttdef"><b>Definition</b> number_of_positive_divisors.cpp:70</div></div>
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<div class="ttc" id="anumber__of__positive__divisors_8cpp_html_ad89ccced8504b5116046cfa03066ffeb"><div class="ttname"><a href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a></div><div class="ttdeci">int number_of_positive_divisors(int n)</div><div class="ttdef"><b>Definition</b> number_of_positive_divisors.cpp:33</div></div>
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</div><!-- fragment --><div class="dynheader">
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Here is the call graph for this function:</div>
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<div class="center"><iframe scrolling="no" frameborder="0" src="../../d0/da2/number__of__positive__divisors_8cpp_ae66f6b31b5ad750f1fe042a706a4e3d4_cgraph.svg" width="362" height="138"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe></div>
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<a id="ad89ccced8504b5116046cfa03066ffeb" name="ad89ccced8504b5116046cfa03066ffeb"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#ad89ccced8504b5116046cfa03066ffeb">◆ </a></span>number_of_positive_divisors()</h2>
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<td class="memname">int number_of_positive_divisors </td>
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<td>(</td>
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<td class="paramtype">int</td> <td class="paramname"><span class="paramname"><em>n</em></span></td><td>)</td>
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<p>Function to compute the number of positive divisors. </p><dl class="params"><dt>Parameters</dt><dd>
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<table class="params">
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<tr><td class="paramname">n</td><td>number to compute divisors for </td></tr>
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</table>
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</dd>
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</dl>
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<dl class="section return"><dt>Returns</dt><dd>number of positive divisors of n (or 1 if n = 0) </dd></dl>
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<div class="fragment"><div class="line"><span class="lineno"> 33</span> {</div>
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<div class="line"><span class="lineno"> 34</span> <span class="keywordflow">if</span> (n < 0) {</div>
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<div class="line"><span class="lineno"> 35</span> n = -n; <span class="comment">// take the absolute value of n</span></div>
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<div class="line"><span class="lineno"> 36</span> }</div>
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<div class="line"><span class="lineno"> 37</span> </div>
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<div class="line"><span class="lineno"> 38</span> <span class="keywordtype">int</span> number_of_divisors = 1;</div>
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<div class="line"><span class="lineno"> 39</span> </div>
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<div class="line"><span class="lineno"> 40</span> <span class="keywordflow">for</span> (<span class="keywordtype">int</span> i = 2; i * i <= n; i++) {</div>
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<div class="line"><span class="lineno"> 41</span> <span class="comment">// This part is doing the prime factorization.</span></div>
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<div class="line"><span class="lineno"> 42</span> <span class="comment">// Note that we cannot find a composite divisor of n unless we would</span></div>
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<div class="line"><span class="lineno"> 43</span> <span class="comment">// already previously find the corresponding prime divisor and dvided</span></div>
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<div class="line"><span class="lineno"> 44</span> <span class="comment">// n by that prime. Therefore, all the divisors found here will</span></div>
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<div class="line"><span class="lineno"> 45</span> <span class="comment">// actually be primes.</span></div>
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<div class="line"><span class="lineno"> 46</span> <span class="comment">// The loop terminates early when it is left with a number n which</span></div>
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<div class="line"><span class="lineno"> 47</span> <span class="comment">// does not have a divisor smaller or equal to sqrt(n) - that means</span></div>
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<div class="line"><span class="lineno"> 48</span> <span class="comment">// the remaining number is a prime itself.</span></div>
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<div class="line"><span class="lineno"> 49</span> <span class="keywordtype">int</span> prime_exponent = 0;</div>
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<div class="line"><span class="lineno"> 50</span> <span class="keywordflow">while</span> (n % i == 0) {</div>
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<div class="line"><span class="lineno"> 51</span> <span class="comment">// Repeatedly divide n by the prime divisor n to compute</span></div>
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<div class="line"><span class="lineno"> 52</span> <span class="comment">// the exponent (e_i in the algorithm description).</span></div>
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<div class="line"><span class="lineno"> 53</span> prime_exponent++;</div>
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<div class="line"><span class="lineno"> 54</span> n /= i;</div>
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<div class="line"><span class="lineno"> 55</span> }</div>
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<div class="line"><span class="lineno"> 56</span> number_of_divisors *= prime_exponent + 1;</div>
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<div class="line"><span class="lineno"> 57</span> }</div>
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<div class="line"><span class="lineno"> 58</span> <span class="keywordflow">if</span> (n > 1) {</div>
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<div class="line"><span class="lineno"> 59</span> <span class="comment">// In case the remaining number n is a prime number itself</span></div>
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<div class="line"><span class="lineno"> 60</span> <span class="comment">// (essentially p_k^1) the final answer is also multiplied by (e_k+1).</span></div>
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<div class="line"><span class="lineno"> 61</span> number_of_divisors *= 2;</div>
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<div class="line"><span class="lineno"> 62</span> }</div>
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<div class="line"><span class="lineno"> 63</span> </div>
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<div class="line"><span class="lineno"> 64</span> <span class="keywordflow">return</span> number_of_divisors;</div>
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<div class="line"><span class="lineno"> 65</span>}</div>
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</div><!-- fragment -->
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<a id="a88ec9ad42717780d6caaff9d3d6977f9" name="a88ec9ad42717780d6caaff9d3d6977f9"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a88ec9ad42717780d6caaff9d3d6977f9">◆ </a></span>tests()</h2>
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<td class="memname">void tests </td>
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<td>(</td>
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<p>Test implementations </p>
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<div class="fragment"><div class="line"><span class="lineno"> 70</span> {</div>
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<div class="line"><span class="lineno"> 71</span> assert(<a class="code hl_function" href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(36) == 9);</div>
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<div class="line"><span class="lineno"> 72</span> assert(<a class="code hl_function" href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(-36) == 9);</div>
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<div class="line"><span class="lineno"> 73</span> assert(<a class="code hl_function" href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(1) == 1);</div>
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<div class="line"><span class="lineno"> 74</span> assert(<a class="code hl_function" href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(2011) == 2); <span class="comment">// 2011 is a prime</span></div>
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<div class="line"><span class="lineno"> 75</span> assert(<a class="code hl_function" href="#ad89ccced8504b5116046cfa03066ffeb">number_of_positive_divisors</a>(756) == 24); <span class="comment">// 756 = 2^2 * 3^3 * 7</span></div>
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<div class="line"><span class="lineno"> 76</span>}</div>
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