From 7fefac0304aa9bfada7711df232a0175d802ec9f Mon Sep 17 00:00:00 2001
From: vishalt12345 <thyagarajanvishal@gmail.com>
Date: Sun, 26 Jun 2022 01:24:18 -0500
Subject: [PATCH 1/3] Added Recursive Extended Euclidean Algorithm

---
 src/algorithms/graph/centroid/centroid.js                  | 0
 .../__test__/extendedEuclideanAlgorithmTest.js             | 0
 .../extendedeuclideanalgorithm.js                          | 7 +++++++
 3 files changed, 7 insertions(+)
 create mode 100644 src/algorithms/graph/centroid/centroid.js
 create mode 100644 src/algorithms/math/extended-euclidean-algorithm/__test__/extendedEuclideanAlgorithmTest.js
 create mode 100644 src/algorithms/math/extended-euclidean-algorithm/extendedeuclideanalgorithm.js

diff --git a/src/algorithms/graph/centroid/centroid.js b/src/algorithms/graph/centroid/centroid.js
new file mode 100644
index 00000000..e69de29b
diff --git a/src/algorithms/math/extended-euclidean-algorithm/__test__/extendedEuclideanAlgorithmTest.js b/src/algorithms/math/extended-euclidean-algorithm/__test__/extendedEuclideanAlgorithmTest.js
new file mode 100644
index 00000000..e69de29b
diff --git a/src/algorithms/math/extended-euclidean-algorithm/extendedeuclideanalgorithm.js b/src/algorithms/math/extended-euclidean-algorithm/extendedeuclideanalgorithm.js
new file mode 100644
index 00000000..08e9e94e
--- /dev/null
+++ b/src/algorithms/math/extended-euclidean-algorithm/extendedeuclideanalgorithm.js
@@ -0,0 +1,7 @@
+export default function extendedEuclideanAlgorithm(a, b){
+    if(b== 0){
+        return [1, 0];
+    }
+    var a = extendedEuclideanAlgorithm(b, a % b);
+    return [a[1], a[0] - Math.floor(a/b) * a[1]];
+}
\ No newline at end of file

From 3b30293c26810c3c4dfc8e15b7153f9008dc79b4 Mon Sep 17 00:00:00 2001
From: vishalt12345 <104476960+vishalt12345@users.noreply.github.com>
Date: Sun, 26 Jun 2022 01:31:51 -0500
Subject: [PATCH 2/3] Create README.md

---
 src/algorithms/math/extended-euclidean-algorithm/README.md | 6 ++++++
 1 file changed, 6 insertions(+)
 create mode 100644 src/algorithms/math/extended-euclidean-algorithm/README.md

diff --git a/src/algorithms/math/extended-euclidean-algorithm/README.md b/src/algorithms/math/extended-euclidean-algorithm/README.md
new file mode 100644
index 00000000..4c60494e
--- /dev/null
+++ b/src/algorithms/math/extended-euclidean-algorithm/README.md
@@ -0,0 +1,6 @@
+<h1>Extended Euclidean Algorithm </h1>
+
+The standard Euclidean algorithm only calculates the gcd of a and b given 2 integers a and b. The extended Euclidean algorithm takes this one step further. For any 2 integers 
+a, b, the extended Euclidean algorithm calculates x and y such that ax + by = gcd(a, b).
+
+

From 5ff34c240ef2773951bcdfe2e674370b876cf660 Mon Sep 17 00:00:00 2001
From: vishalt12345 <104476960+vishalt12345@users.noreply.github.com>
Date: Sun, 26 Jun 2022 01:41:23 -0500
Subject: [PATCH 3/3] Update README.md

---
 .../math/extended-euclidean-algorithm/README.md   | 15 ++++++++++++++-
 1 file changed, 14 insertions(+), 1 deletion(-)

diff --git a/src/algorithms/math/extended-euclidean-algorithm/README.md b/src/algorithms/math/extended-euclidean-algorithm/README.md
index 4c60494e..62cce7a6 100644
--- a/src/algorithms/math/extended-euclidean-algorithm/README.md
+++ b/src/algorithms/math/extended-euclidean-algorithm/README.md
@@ -1,6 +1,19 @@
 <h1>Extended Euclidean Algorithm </h1>
 
 The standard Euclidean algorithm only calculates the gcd of a and b given 2 integers a and b. The extended Euclidean algorithm takes this one step further. For any 2 integers 
-a, b, the extended Euclidean algorithm calculates x and y such that ax + by = gcd(a, b).
+a and b, the extended Euclidean algorithm calculates x and y such that ax + by = gcd(a, b). To perform this, the extended euclidean algorithm makes some slight modifications to the original euclidean algorithm.
 
+The original euclidean algorithm arrived at a = gcd(a,b) and b = 0. Notice that if the original a and b equalled these values, then the solution would merely be x = 1 and y = 0. We can use these 2 values as a starting point for our algorithm. That way, all we would have to do is figure out how x and y change during the transition from (a, b) to (b, a mod b).
 
+Let us say that we have x1 and y1 such that b * x1 + (a mod b) * y1 = gcd(a, b). <br>
+
+Notice that we can substitute a mod b with a - floor(a/b) * b. <br>
+
+Thus, after rearranging the terms, we get gcd(a, b) = a * y1 + b * (x1 - y1 * floor(a / b)). <br>
+
+So after every step, <br>
+
+x = y1 <br>
+y = x1 - y1 * floor(a/b). <br>
+
+**We can use this to eventually backtrack and arrive at the desired x and y.**