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4383_Notes_2o4

# 4383_Notes_2o4 - Math 4383 Section 2.4 Page 1 of 5 Number...

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Math 4383 Section 2.4 Page 1 of 5 Number Theory Chapter 2 Section 2.4 – The Euclidean Algorithm We know that the greatest common divisor of two nonzero integers is always defined, but how do we find it? The Euclidean Algorithm is a systematic way of finding the gcd – and shows us how to write the gcd as a linear combination of the two given numbers. The procedure is found in Euclid’s Elements (although he may not have been the one to come up with it) and so it is called the Euclidean Algorithm. The algorithm is based on the following property of the gcd: Lemma: If a and b are nonzero integers, and a=bq + r with r<b, then gcd(a,b)=gcd(b,r) Proof: The algorithm – Use the division algorithm – 1 1 2 1 2 1 3 2 3 2 4 3 4 2 1 1 1 ... 0 n n n n n n n a q b r b q r r r q r r r q r r r q r r r q r - - - + = + = + = + = + = + = +

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Math 4383 Section 2.4 Page 2 of 5 Then the last nonzero remainder is the gcd of a and b. Furthermore, we can start with the next to the last equation and back substitute to write the gcd as a linear combination of a and b.
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4383_Notes_2o4 - Math 4383 Section 2.4 Page 1 of 5 Number...

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