ΥڡϤƤʥ֥åޡɲΥڡޤϤƤʥ֥åޡ Υڡlivedoor åפɲΥڡޤlivedoor å

pǿΤȤ1/pξŸ

p=25ʳǿΤȤ1/p򾮿ɽΤϽ۴ľȤʤ뤳Ȥ狼äƤ롣ΤȤϡ桼åɤθ߽ˡƱȿؤäƤ롣

ʲǿpͿ줿Ȥƹͤ롣p2,5Ȥ롣ΤȤ1/p򾮿ɽȤ۴ľˤʤäȤơν۴ĹˤĤƹͻ롣

Ūˡ۴ĹdȤơ\frac{1}{p}~=~0.{a_1}{a_2}\cdots{a_{d-1}}{a_d}{a_{d+1}}\cdots{d-1}{a_d}\cdotsȤʤäȤ롣

μξդ10dݤȡ롣

\frac{10^d}{p}~=~{a_1}{a_2}\cdots{a_{d-1}}{a_d}.{a_{d+1}}\cdots{d-1}{a_d}\cdots

κʬȡ롣

\frac{10^d~-~1}{p}={a_1}{a_2}\cdots{a_{d-1}}{a_d}

դʤΤǡդ\frac{10^d~-~1}{p}Ǥ롣10d-1pdzڤ뤳Ȥ̣Τ褦ɽǤ롣

10d1 (mod p)(*)

ʾˤꡢ1/pν۴ĹdǤС(*)ΩĤȤ狼ä

դˡ(*)ΩĤȤˤϡ1/pĹdν۴Ľ۴ľȤɽ뤳Ȥ򼨤

(*)ΩĤȤϡ\frac{10^d~-~1}{p}ʤΤǡ{a_1}{a_2}\cdots{a_{d-1}}{a_d}Τ褦ɽ

\frac{1}{p}
=\frac{10^d~-~1}{p}~\times~{a_1}{a_2}\cdots{a_{d-1}}{a_d}
=\frac{\frac{1}{10^d}}{1-\frac{1}{10^d}}~\times~{a_1}{a_2}\cdots{a_{d-1}}{a_d}
=(\frac{1}{10^d}~+~\frac{1}{10^{2d}}~+~\frac{1}{10^{3d}}~+~\cdots)~\times~{a_1}{a_2}\cdots{a_{d-1}}{a_d}
=\frac{{a_1}{a_2}\cdots{a_{d-1}}{a_d}}{10^d}+~\frac{{a_1}{a_2}\cdots{a_{d-1}}{a_d}}{10^{2d}}+~\frac{{a_1}{a_2}\cdots{a_{d-1}}{a_d}}{10^{3d}}+~\cdots
=0.{a_1}{a_2}\cdots{a_{d-1}}{a_d}{a_{d+1}}\cdots{d-1}{a_d}\cdots

[]p=7ΤȤˡ嵭ѷԤ

p=7ΤȤd=6Ǥʢ1/7=0.142857 142857 142857ġˡ

106-1=9999997ddzڤ롣

\frac{10^6-1}{7}=142857ʤΤǡ\frac{1}{7}=~\frac{142857}{10^6-1}Ȥʤ롣

ǡ\frac{1}{10^6}=0.000001Ǥ뤳ȤȤȡ롣

\frac{1}{10^6-1}
=\frac{\frac{1}{10^6}}{1-\frac{1}{10^6}}
=\frac{0.000001}{1-0.000001}
=0.000001+(0.000001)^2+(0.000001)^3+~\cdots
=0.000001~\,~000001~\,~000001~\cdots

ξդ142857ݤС1/7ξɽ롣

[]1/97ξŸĹ96ν۴ĤȤΤ롣Τˤϡ97ˡȤ10ΰ̿96Ǥ뤳ȡ¨ord_{97}(10)=96Ǥ뤳Ȥ򼨤Ф褤

(97)=96ꡢord_{97}(10)|96Ǥ뤳Ȥ狼롣ʤʤС[̿](a,m)=1Ȥd=ordm(a)֤kˤĤơak1 (mod m)Ȥʤ뤿ɬ׽ʬϡd|kǤ롣פΩĤǤ롣

961,2,3,4,6,8,12,16,24,32,48,96Ǥ롣οǤ10Τ٤97ˡȤƷ׻мΤ褦ˤʤ롣

10^110^210^310^410^610^810^1210^1610^2410^3210^4810^96
103309278150627561961

1096ʳ1˹ƱǤϤʤΤǡ10ΰ̿96Ǥ뤳Ȥ狼롣

ʸ

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