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- Id: 736781
- Posted: 2011-12-14 08:55:50
by danbooru - Size: 1296x1812
- Source: img63.pixiv.net/img/laevateinn495/17211423_big_p0.jpg
- Rating: Safe
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Number 3 is actually a difficult question, so be careful!
Let's practice! [Question] Apply the Euclidean Algorithm to solve the following:
Theorem 3.1 is actually the Euclidean Algorithm. For example, to get (361, 133), 361=133x2+95, therefore (361,133)=(95,133). Similarly, 133=95+38 therefore (95,133)=(95,38), and 95=38x2+19 implies that (95,38)=(19,38)=19, therefore (361,133)=19.
[Theorem 3.1] (a, b) = (a-qb, b) for any integer q. [Proof] Let the GCD of a and b be g. From that, g divides a-qb for any q. Because g is the GCD of a and b, therefore from Theorem 2.1, g is a divisor of (a-qb, b), and therefore g <= (a-qb, b) Let a' = a-qb. Therefore, a = a' + qb, and (a', b) is a divisor of (a, b). Therefore, (a', b) <= g. Combine the two inequalities, and we have (a', b) = g = (a, b), Q.E.D.
Euclidean algorithm
<a href="http://en.wikipedia.org/wiki/Euclidean_algorithm">Euclidean algorithm</a>
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