From 684b6483c1a8f9af199d975f11deda7c6c39821c Mon Sep 17 00:00:00 2001
From: iagorrr04 <iagorrr04@gmail.com>
Date: Sat, 10 Feb 2024 18:53:29 -0300
Subject: [PATCH] :sparkles: adds algorithm to find diophantine equation
 solutions

---
 .../find-solution-diophantine-equation.cpp    | 25 +++++++++++++++++++
 .../find-solution-diophantine-equation.tex    |  5 ++++
 2 files changed, 30 insertions(+)
 create mode 100644 algorithms/math/find-solution-diophantine-equation.cpp
 create mode 100644 algorithms/math/find-solution-diophantine-equation.tex

diff --git a/algorithms/math/find-solution-diophantine-equation.cpp b/algorithms/math/find-solution-diophantine-equation.cpp
new file mode 100644
index 00000000..4640ffaf
--- /dev/null
+++ b/algorithms/math/find-solution-diophantine-equation.cpp
@@ -0,0 +1,25 @@
+ll gcd(ll a, ll b, ll &x, ll &y) {
+  if (b == 0) {
+    x = 1;
+    y = 0;
+    return a;
+  }
+  ll x1, y1;
+  ll d = gcd(b, a % b, x1, y1);
+  x = y1;
+  y = x1 - y1 * (a / b);
+  return d;
+}
+
+bool find_any_solution(ll a, ll b, ll c, ll &x0, ll &y0) {
+  auto g = gcd(a < 0 ? -a : a, b < 0 ? -b : b, x0, y0);
+  if (c % g) {
+    return false;
+  }
+
+  x0 *= c / g;
+  y0 *= c / g;
+  if (a < 0) x0 = -x0;
+  if (b < 0) y0 = -y0;
+  return true;
+}
diff --git a/algorithms/math/find-solution-diophantine-equation.tex b/algorithms/math/find-solution-diophantine-equation.tex
new file mode 100644
index 00000000..38a60b78
--- /dev/null
+++ b/algorithms/math/find-solution-diophantine-equation.tex
@@ -0,0 +1,5 @@
+\subsection{Linear Diophantine Equation: Find any solution}
+
+Given $a$ $b$, $c$ finds the solution to the equation $ax + by = c$, the result will be stored in the reference variables $x0$ and $y0$
+
+time: $O(\log{min(a,b)})$