ExtendedEuclideanAlgorithm (#7172)

* commit1

* Fix clang-format spacing in array declarations

src/main/java/com/thealgorithms/maths/ExtendedEuclideanAlgorithm.java

@@ -0,0 +1,48 @@
+package com.thealgorithms.maths;
+
+/**
+ * In mathematics, the extended Euclidean algorithm is an extension to the
+ * Euclidean algorithm, and computes, in addition to the greatest common divisor
+ * (gcd) of integers a and b, also the coefficients of Bézout's identity, which
+ * are integers x and y such that ax + by = gcd(a, b).
+ *
+ * <p>
+ * For more details, see
+ * https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
+ */
+public final class ExtendedEuclideanAlgorithm {
+
+    private ExtendedEuclideanAlgorithm() {
+    }
+
+    /**
+     * This method implements the extended Euclidean algorithm.
+     *
+     * @param a The first number.
+     * @param b The second number.
+     * @return An array of three integers:
+     *         <ul>
+     *         <li>Index 0: The greatest common divisor (gcd) of a and b.</li>
+     *         <li>Index 1: The value of x in the equation ax + by = gcd(a, b).</li>
+     *         <li>Index 2: The value of y in the equation ax + by = gcd(a, b).</li>
+     *         </ul>
+     */
+    public static long[] extendedGCD(long a, long b) {
+        if (b == 0) {
+            // Base case: gcd(a, 0) = a. The equation is a*1 + 0*0 = a.
+            return new long[] {a, 1, 0};
+        }
+
+        // Recursive call
+        long[] result = extendedGCD(b, a % b);
+        long gcd = result[0];
+        long x1 = result[1];
+        long y1 = result[2];
+
+        // Update coefficients using the results from the recursive call
+        long x = y1;
+        long y = x1 - a / b * y1;
+
+        return new long[] {gcd, x, y};
+    }
+}