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Commit89c7c04

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packagecom.thealgorithms.maths;
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/**
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* In mathematics, the extended Euclidean algorithm is an extension to the
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* Euclidean algorithm, and computes, in addition to the greatest common divisor
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* (gcd) of integers a and b, also the coefficients of Bézout's identity, which
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* are integers x and y such that ax + by = gcd(a, b).
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*
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* <p>
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* For more details, see
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* https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
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*/
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publicfinalclassExtendedEuclideanAlgorithm {
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privateExtendedEuclideanAlgorithm() {
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}
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/**
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* This method implements the extended Euclidean algorithm.
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*
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* @param a The first number.
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* @param b The second number.
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* @return An array of three integers:
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* <ul>
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* <li>Index 0: The greatest common divisor (gcd) of a and b.</li>
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* <li>Index 1: The value of x in the equation ax + by = gcd(a, b).</li>
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* <li>Index 2: The value of y in the equation ax + by = gcd(a, b).</li>
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* </ul>
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*/
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publicstaticlong[]extendedGCD(longa,longb) {
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if (b ==0) {
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// Base case: gcd(a, 0) = a. The equation is a*1 + 0*0 = a.
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returnnewlong[] {a,1,0 };
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}
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// Recursive call
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long[]result =extendedGCD(b,a %b);
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longgcd =result[0];
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longx1 =result[1];
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longy1 =result[2];
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// Update coefficients using the results from the recursive call
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longx =y1;
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longy =x1 -a /b *y1;
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returnnewlong[] {gcd,x,y };
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}
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}
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packagecom.thealgorithms.maths;
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importstaticorg.junit.jupiter.api.Assertions.assertEquals;
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importorg.junit.jupiter.api.Test;
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publicclassExtendedEuclideanAlgorithmTest {
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/**
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* Verifies that the returned values satisfy Bézout's identity: a*x + b*y =
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* gcd(a, b)
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*/
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privatevoidverifyBezoutIdentity(longa,longb,long[]result) {
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longgcd =result[0];
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longx =result[1];
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longy =result[2];
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assertEquals(a *x +b *y,gcd,"Bézout's identity failed for gcd(" +a +", " +b +")");
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}
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@Test
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publicvoidtestExtendedGCD() {
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// Test case 1: General case gcd(30, 50) = 10
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long[]result1 =ExtendedEuclideanAlgorithm.extendedGCD(30,50);
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assertEquals(10,result1[0],"Test Case 1 Failed: gcd(30, 50) should be 10");
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verifyBezoutIdentity(30,50,result1);
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// Test case 2: Another general case gcd(240, 46) = 2
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long[]result2 =ExtendedEuclideanAlgorithm.extendedGCD(240,46);
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assertEquals(2,result2[0],"Test Case 2 Failed: gcd(240, 46) should be 2");
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verifyBezoutIdentity(240,46,result2);
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// Test case 3: Base case where b is 0, gcd(10, 0) = 10
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long[]result3 =ExtendedEuclideanAlgorithm.extendedGCD(10,0);
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assertEquals(10,result3[0],"Test Case 3 Failed: gcd(10, 0) should be 10");
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verifyBezoutIdentity(10,0,result3);
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// Test case 4: Numbers are co-prime gcd(17, 13) = 1
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long[]result4 =ExtendedEuclideanAlgorithm.extendedGCD(17,13);
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assertEquals(1,result4[0],"Test Case 4 Failed: gcd(17, 13) should be 1");
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verifyBezoutIdentity(17,13,result4);
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// Test case 5: One number is a multiple of the other gcd(100, 20) = 20
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long[]result5 =ExtendedEuclideanAlgorithm.extendedGCD(100,20);
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assertEquals(20,result5[0],"Test Case 5 Failed: gcd(100, 20) should be 20");
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verifyBezoutIdentity(100,20,result5);
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}
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}

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