P
  分數如果表成---  (N,P)=1     若將分數化為小數,循環節最多N-1位
               N


  請問證明如何寫會比較清楚明瞭?



作者: JohnMash (Paul) 看板: Math
※ 引述《gary85238 (小黃)》之銘言：
: Let a,b 屬於 N , a<b and a is relative prime to b.
: Use the Pigeonhole Principle to show that the decimal representation
: of the fraction a/b either terminate or repeats.

if a/b terminates in its representation, done.

if a/b does not terminate


consider a, 10a, 100a, ....,10^b a,....

        a=a_0 (mod b)

      10a=a_1 (mod b)

      100a=a_2 (mod b)

      ......

because a_0,a_1,a_2,...,a_b are all less than b

  then there exist at least two a_i, a_j equal

   that is 10^i a= kb+10^j a

    10^i a/b=k+10^j a/b

  that is 10^i a/b has the same decimal part as 10^j a/b

  it means a/b repeats its decimal representation

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