Math 4383
Section 2.4
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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 –
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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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 Spring '09
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 Number Theory, Integers, Greatest common divisor, Euclidean algorithm, gcd, nonzero integers

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