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The extended euclidean algorithm, a method used to find the greatest common divisor (gcd) of two numbers and determine their linear combination. Two examples are provided, one with m = 65 and n = 40, and the other with m = 1239 and n = 735.
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Example 1: m = 65, n = 40
Step 1: The (usual) Euclidean algorithm:
(1) 65 = 1 · 40 + 25 (2) 40 = 1 · 25 + 15 (3) 25 = 1 · 15 + 10 (4) 15 = 1 · 10 + 5 10 = 2 · 5
Therefore: gcd(65, 40) = 5.
Step 2: Using the method of back-substitution:
5 (4) = 15 − 10 (3) = 15 − (25 − 15) = 2 · 15 − 25 (2) = 2(40 − 25) − 25 = 2 · 40 − 3 · 25 (1) = 2 · 40 − 3(65 − 40) = 5 · 40 − 3 · 65
Conclusion: 65(︸ −︷︷3) ︸ x
y
Example 2: m = 1239, n = 735
Step 1: The (usual) Euclidean algorithm:
(1) 1239 = 1 · 735 + 504 (2) 735 = 1 · 504 + 231 (3) 504 = 2 · 231 + 42 (4) 231 = 5 · 42 + 21 42 = 2 · 21
Therefore: gcd(1239, 735) = 21.
Step 2: Using the method of back-substitution:
21 (4) = 231 − 5 · 42 (3) = 231 − 5(504 − 2 · 231) = 11 · 231 − 5 · 504 (2) = 11(735 − 504) − 5 · 504 = 11 · 735 − 16 · 504 (1) = 11 · 735 − 16(1239 − 735) = 27 · 735 − 16 · 1239
Conclusion: 1239(︸ −︷︷16) ︸ x
y
