The Euclidean Algorithm

Discrete Math · Axiom Academy

EXAMPLE The Euclidean Algorithm Finding the GCD and discovering Bézout coefficients through repeated division Excellent work! You've mastered the Euclidean Algorithm and its extension. Here's what we learned: Euclidean Algorithm: Repeatedly apply the division algorithm until the remainder is 0. The last non-zero remainder is the GCD. Efficiency: The algorithm is remarkably efficient, reducing large numbers to the GCD in just a few steps. Extended Algorithm: By working backwards through our divisions, we can express the GCD as a linear combination: gcd(a,b) = ax + by. Bézout's Identity: For any integers a and b, there exist integers x and y (called Bézout coefficients) such that ax + by = gcd(a,b). Applications: This algorithm is fundamental to number theory, cryptography (RSA), and solving linear Diophantine equations. The Euclidean Algorithm, dating back to ancient Greece, remains one of the most elegant and useful algorithms in all of mathematics. Its efficiency and versatility make it indispensable in modern applications!

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