<div>Bezout
was a famous mathematician who discovered many beautiful formulas. One of his
most famous theorem/identities dealt with numbers and their greatest common
factors. His theorem states, if integers <span class="ltx_Math" contenteditable="false" data-equation="a">\(a\)</span> and <span class="ltx_Math" contenteditable="false" data-equation="b">\(b\)</span> are relatively prime, then
there exists <span class="ltx_Math" contenteditable="false" data-equation="x">\(x\)</span> and <span class="ltx_Math" contenteditable="false" data-equation="y">\(y\)</span>,  integers to satisfy the equation <span class="ltx_Math" contenteditable="false" data-equation="ax+by=1">\(ax+by=1\)</span>. For any
integers <span class="ltx_Math" contenteditable="false" data-equation="a">\(a\)</span> and <span class="ltx_Math" contenteditable="false" data-equation="b">\(b\)</span> excluding zero, let <span class="ltx_Math" contenteditable="false" data-equation="d">\(d\)</span> be the greatest common divisor such that
<span class="ltx_Math" contenteditable="false" data-equation="d">\(d\)</span> equals the gcd of <span class="ltx_Math" contenteditable="false" data-equation="a">\(a\)</span> and <span class="ltx_Math" contenteditable="false" data-equation="b">\(b\)</span> in other words, <span class="ltx_Math" contenteditable="false" data-equation="d">\(d\)</span><span class="ltx_Math" contenteditable="false" data-equation="=">\(=\)</span>gcd<span class="ltx_Math" contenteditable="false" data-equation="\left(a,b\right)">\(\left(a,b\right)\)</span>. If this is true then
there must exist integers known as <span class="ltx_Math" contenteditable="false" data-equation="x">\(x\)</span> and <span class="ltx_Math" contenteditable="false" data-equation="y">\(y\)</span>&nbsp;to satisfy the equation <span class="ltx_Math" contenteditable="false" data-equation="ax+by=d">\(ax+by=d\)</span>. <br></div>