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자유게시판
수학을 주제로 떠들어 보세요!
[수학 정보] 유클리드 호제법
mathwizard 2020.12.13 05:47 조회 683
공유하기
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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    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: is_dir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 201

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 201
    Function: is_dir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: mkdir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 202

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 202
    Function: mkdir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_exists(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/639bee393eecbc62256936a8e64d17b1.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 212

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 212
    Function: file_exists

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/639bee393eecbc62256936a8e64d17b1.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(/DATA/upload/polymath/latex/639bee393eecbc62256936a8e64d17b1.gif): Failed to open stream: Operation not permitted

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: is_dir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 201

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 201
    Function: is_dir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: mkdir(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 202

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 202
    Function: mkdir

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_exists(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/486d654608b39e7d01bf9371c04dd09d.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 212

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 212
    Function: file_exists

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(): open_basedir restriction in effect. File(/DATA/upload/polymath/latex/486d654608b39e7d01bf9371c04dd09d.gif) is not within the allowed path(s): (/volume1/web/PhpstormProjects/www_polymath_co_kr:/volume1/web:/tmp:/var/tmp:/var/services/tmp:/var/services/web:/volume1/@tmp)

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    A PHP Error was encountered

    Severity: Warning

    Message: file_put_contents(/DATA/upload/polymath/latex/486d654608b39e7d01bf9371c04dd09d.gif): Failed to open stream: Operation not permitted

    Filename: libraries/Common.php

    Line Number: 213

    Backtrace:

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 213
    Function: file_put_contents

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
    Line: 236
    Function: getLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view_comment_list.php
    Line: 90
    Function: parseLatexImg

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/views/ver3/inc/view.php
    Line: 343
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
    Line: 558
    Function: view

    File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
    Line: 315
    Function: require_once

    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

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property Article_model::$best_cnt is deprecated

Filename: ver3/Article_model.php

Line Number: 18

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Article_model.php
Line: 18
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 322
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 392
Function: getFloatingMyInfo

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 559
Function: get_footer

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

A PHP Error was encountered

Severity: 8192

Message: Creation of dynamic property Article_model::$popular_cnt is deprecated

Filename: ver3/Article_model.php

Line Number: 19

Backtrace:

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/models/ver3/Article_model.php
Line: 19
Function: _error_handler

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 322
Function: model

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/libraries/Common.php
Line: 392
Function: getFloatingMyInfo

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/application/controllers/ver3/Contents.php
Line: 559
Function: get_footer

File: /volume1/web/PhpstormProjects/www_polymath_co_kr/index.php
Line: 315
Function: require_once

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