php-doc-en/reference/gmp/functions/gmp-gcdext.xml

<?xml version="1.0" encoding="utf-8"?>
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<refentry xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://docbook.org/ns/docbook" xml:id="function.gmp-gcdext">
 <refnamediv>
  <refname>gmp_gcdext</refname>
  <refpurpose>Calculate GCD and multipliers</refpurpose>
 </refnamediv>
 
 <refsect1 role="description">
  &reftitle.description;
  <methodsynopsis>
   <type>array</type><methodname>gmp_gcdext</methodname>
   <methodparam><type>resource</type><parameter>a</parameter></methodparam>
   <methodparam><type>resource</type><parameter>b</parameter></methodparam>
  </methodsynopsis>
  <para>
   Calculates g, s, and t, such that <literal>a*s + b*t = g =
   gcd(a,b)</literal>, where gcd is the greatest common divisor. Returns
   an array with respective elements g, s and t.
  </para>
  <para>
   This function can be used to solve linear Diophantine equations in two
   variables. These are equations that allow only integer solutions and have the form:
   <literal>a*x + b*y = c</literal>.
   For more information, go to the <link xlink:href="&url.diophantine-equation;">"Diophantine
   Equation" page at MathWorld</link>
  </para>
 </refsect1>

 <refsect1 role="parameters">
  &reftitle.parameters;
  <para>
   <variablelist>
    <varlistentry>
     <term><parameter>a</parameter></term>
     <listitem>
      &gmp.parameter; 
     </listitem>
    </varlistentry>
    <varlistentry>
     <term><parameter>b</parameter></term>
     <listitem>
      &gmp.parameter; 
     </listitem>
    </varlistentry>
   </variablelist>
  </para>
 </refsect1>

 <refsect1 role="returnvalues">
  &reftitle.returnvalues;
  <para>
   An <type>array</type> of GMP numbers.
  </para>
 </refsect1>

 <refsect1 role="examples">
  &reftitle.examples;
  <para>
   <example>
    <title>Solving a linear Diophantine equation</title>
    <programlisting role="php">
<![CDATA[
<?php
// Solve the equation a*s + b*t = g
// where a = 12, b = 21, g = gcd(12, 21) = 3
$a = gmp_init(12);
$b = gmp_init(21);
$g = gmp_gcd($a, $b);
$r = gmp_gcdext($a, $b);

$check_gcd = (gmp_strval($g) == gmp_strval($r['g']));
$eq_res = gmp_add(gmp_mul($a, $r['s']), gmp_mul($b, $r['t']));
$check_res = (gmp_strval($g) == gmp_strval($eq_res));

if ($check_gcd && $check_res) {
    $fmt = "Solution: %d*%d + %d*%d = %d\n";
    printf($fmt, gmp_strval($a), gmp_strval($r['s']), gmp_strval($b),
    gmp_strval($r['t']), gmp_strval($r['g']));
} else {
    echo "Error while solving the equation\n";
}

// output: Solution: 12*2 + 21*-1 = 3
?>
]]>
    </programlisting>
   </example>
  </para>
 </refsect1>
</refentry>

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