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<div class="contents">
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<p>An algorithm to divide two numbers under modulo p <a href="https://www.geeksforgeeks.org/modular-division">Modular Division</a>
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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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Include dependency graph for modular_division.cpp:</div>
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<tr class="memitem:dd/d47/namespacemath"><td class="memItemLeft" align="right" valign="top">  </td><td class="memItemRight" valign="bottom"><a class="el" href="../../dd/d47/namespacemath.html">math</a></td></tr>
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<tr class="memdesc:dd/d47/namespacemath"><td class="mdescLeft"> </td><td class="mdescRight">for <a class="elRef" target="_blank" href="http://en.cppreference.com/w/cpp/container/vector.html">std::vector</a> <br /></td></tr>
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<tr class="separator:"><td class="memSeparator" colspan="2"> </td></tr>
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<tr class="memitem:d1/d64/namespacemodular__division"><td class="memItemLeft" align="right" valign="top">  </td><td class="memItemRight" valign="bottom"><a class="el" href="../../d1/d64/namespacemodular__division.html">modular_division</a></td></tr>
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<tr class="memdesc:d1/d64/namespacemodular__division"><td class="mdescLeft"> </td><td class="mdescRight">Functions for <a href="https://www.geeksforgeeks.org/modular-division">Modular Division</a> implementation. <br /></td></tr>
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Functions</h2></td></tr>
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<tr class="memitem:a66cdf93153cbd1408bd74ac68961d179"><td class="memItemLeft" align="right" valign="top">uint64_t </td><td class="memItemRight" valign="bottom"><a class="el" href="../../df/d72/modular__division_8cpp.html#a66cdf93153cbd1408bd74ac68961d179">math::modular_division::power</a> (uint64_t a, uint64_t b, uint64_t c)</td></tr>
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<tr class="memdesc:a66cdf93153cbd1408bd74ac68961d179"><td class="mdescLeft"> </td><td class="mdescRight">This function calculates a raised to exponent b under modulo c using modular exponentiation. <a href="../../df/d72/modular__division_8cpp.html#a66cdf93153cbd1408bd74ac68961d179">More...</a><br /></td></tr>
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<tr class="memitem:a905e368ae121beb7e7ea35349ddcdac7"><td class="memItemLeft" align="right" valign="top">uint64_t </td><td class="memItemRight" valign="bottom"><a class="el" href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a> (uint64_t a, uint64_t b, uint64_t p)</td></tr>
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<tr class="memdesc:a905e368ae121beb7e7ea35349ddcdac7"><td class="mdescLeft"> </td><td class="mdescRight">This function calculates modular division. <a href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">More...</a><br /></td></tr>
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<tr class="memitem:aa8dca7b867074164d5f45b0f3851269d"><td class="memItemLeft" align="right" valign="top">static void </td><td class="memItemRight" valign="bottom"><a class="el" href="../../df/d72/modular__division_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a> ()</td></tr>
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<tr class="memitem:a0ddf1224851353fc92bfbff6f499fa97"><td class="memItemLeft" align="right" valign="top">int </td><td class="memItemRight" valign="bottom"><a class="el" href="../../df/d72/modular__division_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97">main</a> (int argc, char *argv[])</td></tr>
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<tr class="memdesc:a0ddf1224851353fc92bfbff6f499fa97"><td class="mdescLeft"> </td><td class="mdescRight">Main function. <a href="../../df/d72/modular__division_8cpp.html#a0ddf1224851353fc92bfbff6f499fa97">More...</a><br /></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>An algorithm to divide two numbers under modulo p <a href="https://www.geeksforgeeks.org/modular-division">Modular Division</a> </p>
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<p>To calculate division of two numbers under modulo p Modulo operator is not distributive under division, therefore we first have to calculate the inverse of divisor using <a href="https://en.wikipedia.org/wiki/Fermat%27s_little_theorem">Fermat's little theorem</a> Now, we can multiply the dividend with the inverse of divisor and modulo is distributive over multiplication operation. Let, We have 3 numbers a, b, p To compute (a/b)p (a/b)p ≡ (a*(inverse(b)))p ≡ ((ap)*inverse(b)p)p NOTE: For the existence of inverse of 'b', 'b' and 'p' must be coprime For simplicity we take p as prime Time Complexity: O(log(b)) Example: ( 24 / 3 ) % 5 => 8 % 5 = 3 — (i) Now the inverse of 3 is 2 (24 * 2) % 5 = (24 % 5) * (2 % 5) = (4 * 2) % 5 = 3 — (ii) (i) and (ii) are equal hence the answer is correct. </p><dl class="section see"><dt>See also</dt><dd><a class="el" href="../../d8/d53/modular__inverse__fermat__little__theorem_8cpp.html" title="C++ Program to find the modular inverse using Fermat's Little Theorem">modular_inverse_fermat_little_theorem.cpp</a>, <a class="el" href="../../d0/d6d/modular__exponentiation_8cpp.html" title="C++ Program for Modular Exponentiation Iteratively.">modular_exponentiation.cpp</a> </dd></dl>
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<dl class="section author"><dt>Author</dt><dd><a href="https://github.com/shubhamamsa">Shubham Yadav</a> </dd></dl>
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</div><h2 class="groupheader">Function Documentation</h2>
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<a id="a0ddf1224851353fc92bfbff6f499fa97"></a>
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<h2 class="memtitle"><span class="permalink"><a href="#a0ddf1224851353fc92bfbff6f499fa97">◆ </a></span>main()</h2>
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<td class="memname">int main </td>
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<td>(</td>
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<td class="paramtype">int </td>
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<td>)</td>
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<p>Main function. </p>
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<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">argc</td><td>commandline argument count (ignored) </td></tr>
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<tr><td class="paramname">argv</td><td>commandline array of arguments (ignored) </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>0 on exit </dd></dl>
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<div class="fragment"><div class="line"><a name="l00112"></a><span class="lineno"> 112</span>  {</div>
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<div class="line"><a name="l00113"></a><span class="lineno"> 113</span>  <a class="code" href="../../df/d72/modular__division_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a>(); <span class="comment">// execute the tests</span></div>
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<div class="line"><a name="l00114"></a><span class="lineno"> 114</span>  <span class="keywordflow">return</span> 0;</div>
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<div class="line"><a name="l00115"></a><span class="lineno"> 115</span> }</div>
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<div class="ttc" id="amodular__division_8cpp_html_aa8dca7b867074164d5f45b0f3851269d"><div class="ttname"><a href="../../df/d72/modular__division_8cpp.html#aa8dca7b867074164d5f45b0f3851269d">test</a></div><div class="ttdeci">static void test()</div><div class="ttdef"><b>Definition:</b> modular_division.cpp:88</div></div>
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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="../../df/d72/modular__division_8cpp_a0ddf1224851353fc92bfbff6f499fa97_cgraph.svg" width="274" height="38"><p><b>This browser is not able to show SVG: try Firefox, Chrome, Safari, or Opera instead.</b></p></iframe>
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<h2 class="memtitle"><span class="permalink"><a href="#a905e368ae121beb7e7ea35349ddcdac7">◆ </a></span>mod_division()</h2>
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<td class="memname">uint64_t math::modular_division::mod_division </td>
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<p>This function calculates modular division. </p>
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<dl class="params"><dt>Parameters</dt><dd>
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<tr><td class="paramname">a</td><td>integer dividend </td></tr>
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<tr><td class="paramname">b</td><td>integer divisor </td></tr>
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<tr><td class="paramname">p</td><td>integer modulo </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>a/b modulo c </dd></dl>
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<p>Calculate the inverse of b</p>
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<p>Calculate the final result</p>
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<div class="fragment"><div class="line"><a name="l00074"></a><span class="lineno"> 74</span>  {</div>
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<div class="line"><a name="l00075"></a><span class="lineno"> 75</span>  uint64_t inverse = <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#a7cf5feaf168b88e74544da59ed830311">power</a>(b, p - 2, p) % p; <span class="comment">/// Calculate the inverse of b</span></div>
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<div class="line"><a name="l00076"></a><span class="lineno"> 76</span>  uint64_t <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a> =</div>
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<div class="line"><a name="l00077"></a><span class="lineno"> 77</span>  ((a % p) * (inverse % p)) % p; <span class="comment">/// Calculate the final result</span></div>
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<div class="line"><a name="l00078"></a><span class="lineno"> 78</span>  <span class="keywordflow">return</span> <a class="code" href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">result</a>;</div>
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<div class="line"><a name="l00079"></a><span class="lineno"> 79</span> }</div>
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<div class="ttc" id="afibonacci__sum_8cpp_html_a7cf5feaf168b88e74544da59ed830311"><div class="ttname"><a href="../../de/dc3/fibonacci__sum_8cpp.html#a7cf5feaf168b88e74544da59ed830311">math::fibonacci_sum::power</a></div><div class="ttdeci">math::fibonacci_sum::matrix power(math::fibonacci_sum::matrix T, uint64_t ex)</div><div class="ttdef"><b>Definition:</b> fibonacci_sum.cpp:57</div></div>
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<div class="ttc" id="afibonacci__sum_8cpp_html_aadb40ac4c74a7efc0680b83eeee138aa"><div class="ttname"><a href="../../de/dc3/fibonacci__sum_8cpp.html#aadb40ac4c74a7efc0680b83eeee138aa">math::fibonacci_sum::result</a></div><div class="ttdeci">uint64_t result(uint64_t n)</div><div class="ttdef"><b>Definition:</b> fibonacci_sum.cpp:76</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#a66cdf93153cbd1408bd74ac68961d179">◆ </a></span>power()</h2>
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<td class="memname">uint64_t math::modular_division::power </td>
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<td>(</td>
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<td class="paramtype">uint64_t </td>
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<td class="paramname"><em>c</em> </td>
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<p>This function calculates a raised to exponent b under modulo c using modular exponentiation. </p>
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<tr><td class="paramname">a</td><td>integer base </td></tr>
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<tr><td class="paramname">b</td><td>unsigned integer exponent </td></tr>
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<tr><td class="paramname">c</td><td>integer modulo </td></tr>
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<dl class="section return"><dt>Returns</dt><dd>a raised to power b modulo c </dd></dl>
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<p>Initialize the answer to be returned</p>
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<p>Update a if it is more than or equal to c</p>
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<p>In case a is divisible by c;</p>
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<p>If b is odd, multiply a with answer</p>
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<p>b must be even now</p>
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<p>b = b/2</p>
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<div class="fragment"><div class="line"><a name="l00049"></a><span class="lineno"> 49</span>  {</div>
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<div class="line"><a name="l00050"></a><span class="lineno"> 50</span>  uint64_t <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#ad8389ed58fd0ec66df248014775ad1fa">ans</a> = 1; <span class="comment">/// Initialize the answer to be returned</span></div>
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<div class="line"><a name="l00051"></a><span class="lineno"> 51</span>  a = a % c; <span class="comment">/// Update a if it is more than or equal to c</span></div>
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<div class="line"><a name="l00052"></a><span class="lineno"> 52</span>  <span class="keywordflow">if</span> (a == 0) {</div>
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<div class="line"><a name="l00053"></a><span class="lineno"> 53</span>  <span class="keywordflow">return</span> 0; <span class="comment">/// In case a is divisible by c;</span></div>
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<div class="line"><a name="l00054"></a><span class="lineno"> 54</span>  }</div>
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<div class="line"><a name="l00055"></a><span class="lineno"> 55</span>  <span class="keywordflow">while</span> (b > 0) {<span class="comment"></span></div>
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<div class="line"><a name="l00056"></a><span class="lineno"> 56</span> <span class="comment"> /// If b is odd, multiply a with answer</span></div>
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<div class="line"><a name="l00057"></a><span class="lineno"> 57</span> <span class="comment"></span> <span class="keywordflow">if</span> (b & 1) {</div>
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<div class="line"><a name="l00058"></a><span class="lineno"> 58</span>  <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#ad8389ed58fd0ec66df248014775ad1fa">ans</a> = ((<a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#ad8389ed58fd0ec66df248014775ad1fa">ans</a> % c) * (a % c)) % c;</div>
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<div class="line"><a name="l00059"></a><span class="lineno"> 59</span>  }<span class="comment"></span></div>
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<div class="line"><a name="l00060"></a><span class="lineno"> 60</span> <span class="comment"> /// b must be even now</span></div>
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<div class="line"><a name="l00061"></a><span class="lineno"> 61</span> <span class="comment"></span> b = b >> 1; <span class="comment">/// b = b/2</span></div>
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<div class="line"><a name="l00062"></a><span class="lineno"> 62</span>  a = ((a % c) * (a % c)) % c;</div>
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<div class="line"><a name="l00063"></a><span class="lineno"> 63</span>  }</div>
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<div class="line"><a name="l00064"></a><span class="lineno"> 64</span>  <span class="keywordflow">return</span> <a class="code" href="../../d7/d35/matrix__exponentiation_8cpp.html#ad8389ed58fd0ec66df248014775ad1fa">ans</a>;</div>
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<div class="line"><a name="l00065"></a><span class="lineno"> 65</span> }</div>
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<div class="ttc" id="amatrix__exponentiation_8cpp_html_ad8389ed58fd0ec66df248014775ad1fa"><div class="ttname"><a href="../../d7/d35/matrix__exponentiation_8cpp.html#ad8389ed58fd0ec66df248014775ad1fa">ans</a></div><div class="ttdeci">ll ans(ll n)</div><div class="ttdef"><b>Definition:</b> matrix_exponentiation.cpp:91</div></div>
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<h2 class="memtitle"><span class="permalink"><a href="#aa8dca7b867074164d5f45b0f3851269d">◆ </a></span>test()</h2>
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<p>Function for testing power function. test cases and assert statement. </p><dl class="section return"><dt>Returns</dt><dd><code>void</code> </dd></dl>
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<div class="fragment"><div class="line"><a name="l00088"></a><span class="lineno"> 88</span>  {</div>
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<div class="line"><a name="l00089"></a><span class="lineno"> 89</span>  uint64_t test_case_1 = <a class="code" href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(8, 2, 2);</div>
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<div class="line"><a name="l00090"></a><span class="lineno"> 90</span>  assert(test_case_1 == 0);</div>
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<div class="line"><a name="l00091"></a><span class="lineno"> 91</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 1 Passed!"</span> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00092"></a><span class="lineno"> 92</span>  uint64_t test_case_2 = <a class="code" href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(15, 3, 7);</div>
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<div class="line"><a name="l00093"></a><span class="lineno"> 93</span>  assert(test_case_2 == 5);</div>
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<div class="line"><a name="l00094"></a><span class="lineno"> 94</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 2 Passed!"</span> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00095"></a><span class="lineno"> 95</span>  uint64_t test_case_3 = <a class="code" href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(10, 5, 2);</div>
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<div class="line"><a name="l00096"></a><span class="lineno"> 96</span>  assert(test_case_3 == 0);</div>
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<div class="line"><a name="l00097"></a><span class="lineno"> 97</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 3 Passed!"</span> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00098"></a><span class="lineno"> 98</span>  uint64_t test_case_4 = <a class="code" href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(81, 3, 5);</div>
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<div class="line"><a name="l00099"></a><span class="lineno"> 99</span>  assert(test_case_4 == 2);</div>
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<div class="line"><a name="l00100"></a><span class="lineno"> 100</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 4 Passed!"</span> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00101"></a><span class="lineno"> 101</span>  uint64_t test_case_5 = <a class="code" href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a>(12848, 73, 29);</div>
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<div class="line"><a name="l00102"></a><span class="lineno"> 102</span>  assert(test_case_5 == 2);</div>
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<div class="line"><a name="l00103"></a><span class="lineno"> 103</span>  <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/basic_ostream.html">std::cout</a> << <span class="stringliteral">"Test 5 Passed!"</span> << <a class="codeRef" target="_blank" href="http://en.cppreference.com/w/cpp/io/manip/endl.html">std::endl</a>;</div>
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<div class="line"><a name="l00104"></a><span class="lineno"> 104</span> }</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="amodular__division_8cpp_html_a905e368ae121beb7e7ea35349ddcdac7"><div class="ttname"><a href="../../df/d72/modular__division_8cpp.html#a905e368ae121beb7e7ea35349ddcdac7">math::modular_division::mod_division</a></div><div class="ttdeci">uint64_t mod_division(uint64_t a, uint64_t b, uint64_t p)</div><div class="ttdoc">This function calculates modular division.</div><div class="ttdef"><b>Definition:</b> modular_division.cpp:74</div></div>
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