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[수학 정보] 유클리드 호제법
mathwizard 2020.12.13 05:47 조회 659
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페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

책을 읽다가 유클리드 호제법이라는 것에 대해 알게 되었습니다.

유클리드 호제법은 어떤 두 수의 최대공약수를 구하는 방법 중 하나로,

선뜻 나누어떨어지지 않는 두 수에서 큰 수를 작은 수로 나누어서

나온 나머지로 다시 앞의 작은 수를 나누는 계산을 되풀이하는 방법입니다.

58과 46을 예시로 들어 계산해보겠습니다.

58과 46중에서 58이 더 크니 58을 46으로 나누어야 합니다.

58을 46으로 나누면 1번 들어가고, 12가 남으므로 식으로 나타내면

58=46x1+12 입니다.

다시 46을 12로 나누면 3번 들어가고 10이 남으므로

46=12x3+10이 됩니다. 같은 방법으로 게속 계산한 식을 56나누기46부터 써보면

 

58=46x1+12

46=12x3+10

12=10x1+2

10=2x5+0

 

이 됩니다.

따라서 58과 46의 최대공약수는 0이 아닌 마지막 나머지 2입니다.

 

최소공배수는 이와 비슷한 방법으로 풀 수 없을까요?

 

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  •  
    아인수타인 Lv.12 2020.12.13 07:43

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    사실 최소공배수는 최대공약수만 알면 바로 풀 수 있습니다. 예를 들어 처음 두 수가 A,B이고 최대공약수를 g라고 하면, A=ga, B=gb, \textup{gcd}(a,b)=1입니다. 이 때, 두 수의 최소공배수는 gab가 되는데, AB=ga\times gb=g\times gab=\textup{gcd}(A,B)\times \textup{lcm}(A,B)가 되므로, 두 수의 최소공배수는 원래 두 수를 곱하고, 최대공약수로 나눠 주면 됩니다.

    댓글 작성하기 좋아요0 댓글수1
    •  
      mathwizard Lv.7 2020.12.13 17:30

      와 그건 몰랐네요.

      고맙습니다!

      좋아요0
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