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수학동아 - 폴리매스
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함께 풀고 싶은 문제
창의력을 기를 수 있는 수학 문제 또는 퍼즐을 내는 곳입니다.
[창의 퍼즐]
sincostan 2019.09.15 04:21 조회 570
문제 푼 사람 0명
공유하기
페이스북 트위터 카카오톡 카카오스토리 네이버블로그 네이버밴드

N!이 1+2+3+...+N으로 나누어떨어지도록 하는 양의 정수 N을 모두 구하여라.

이 문제 어떠셨나요?

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  •  
    디듀우 Lv.7 2019.09.15 06:09

    N(N+1)/2=1*2*3*...*N

    일단 1, 3일때 성립하는 것은 대입해보면 알 수 있습니다. 그 이후론 좌변과 우변 차가 커져서 더 없을 것 같긴 한데... 엄밀한 증명이 필요하겠네요.

     

    댓글 작성하기 좋아요0 댓글수2
    •  
      디듀우 Lv.7 2019.09.15 06:19

      4=N일 때 N(N+1)/2<N!

      k=N일 때 N(N+1)/2<N! 이라면

      k+1=N일 때는 N(N+1)/2=(k+1)(k+2)/2={k(k+1)/2}*{(k+2)/k}, N!=(k+1)k!이다. (k+2)/k>k+1은 k>0일 때 2<k^2와 동치이고 이는 k>4일 때 성립하므로 N(N+1)/2<N

      따라서 N>=4일 때는 N(N+1)/2<N!, 3보다 큰 N(N+1)/2=N!의 해는 존재하지 않는다.

      좋아요0
    •  
      디듀우 Lv.7 2019.09.15 06:19

      악! 나누어떨어지도록이었네요... 문제를 이제야 제대로 보다니...

      좋아요0
  •  
    바람개비 Lv.5 2019.09.15 07:21

    다음 수가 소수가 아닌 자연수는 다 되지 않을까요? (물론 1은 포함)

    댓글 작성하기 좋아요0 댓글수2
    •  
      디듀우 Lv.7 2019.09.15 07:41

      1 됩니다! 둘이 같아요!

      좋아요0
    •  
      바람개비 Lv.5 2019.09.15 07:53

      잘못 썼어요:(

      좋아요0
  •  
    code Lv.5 2019.09.16 06:08

    2*N!/(N+1)N=2*(N-1)!/(N+1).

    N+1을 소수로 만드는 N이아니면 모두가능하다(1 제외).

    2를 제외한 소수분모로는 나누어 떨어지지않는다.

    따라서 다음 수가 소수인 N(1제외)은 불가능하다.

    또한 합성수의 경우 나누어떨어지지않기위해서는 N이 소수이고 N+1의 약수여야한다.

    그런 경우는 없다.

    댓글 작성하기 좋아요0 댓글수1
    •  
      sincostan Lv.7 2019.09.16 08:07

      code님 정답입니다.

      좋아요0
  •  
    sincostan Lv.7 2019.10.03 10:05

    이 문제를 낸게 몇일 전인데 벌써 엄청 밀려있네

    댓글 작성하기 좋아요0 댓글수0
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