Ȥ
ǥľǤ

(C1) 666666666666666666666666666666*666666666666666666666666666668;
(D1) 444444444444444444444444444444 888888888888888888888888888888
(C2) 4*30+8*30;
(D2) 360
˭ԡ 525ڡ 0:05:49 MAIL:fttnm528@ybb.ne.jp 27461
ޥ
äȴñޤ...m(__)m
PowerBook G4 525ڡ 0:08:15 MAIL:masaru-y@sansu.org HomePage:27462
LiE
666*668=444888ǵĤޤ
4*30=120
8*30=240
120+240=360
Իļˡ 525ڡ 0:08:25 MAIL:njswg145@ybb.ne.jp 27463
fumio
Фϡ
525ڡ 0:08:46 27464

*ʤƥ*򤹤ȳڤäƤȤǤΤʡ
äȴޤ
525ڡ 0:09:37 27465
äѥȥ
6*8=48
66*68=4488
666*668=444888
䤷ޤ
525ڡ 0:10:00 27466
DrK
6668=22368ͤз׻ڤǤ

222000000000000000000000000000004Ȥʤꡢ4488Ȥʤޤ
430+830=360
ǤϲΤ240äƤޤޤ
Τ֤̤Ȥʤޤ
2/32/3=4/9ȤʤȹͤƤƻפĤޤ
٤Ͼγڱࡡ 525ڡ 0:11:28 MAIL:satoka@star.odn.ne.jp 27467
ڼ
׻Ǥ
666686=400008
夺Ĥ餷ʤƬ)30¤٤ޤ
͡ 525ڡ 0:11:59 MAIL:iida@ae.keio.ac.jp 27468
Ҥ
ޤˤϡʤΤȥۥäȤޤ
ߣ
ߣʤɤƱǤ
525ڡ 0:15:10 27469
㡼ߡ
#27468 ʤۤɡΤȤ

n Ĥξ硤 = (66667)^2-1 = (2*10^n +1)^2 /9 -1
= 4*(10^{2n}-1)/9 + 4*(10^n -1)/9
ѷƤޤ
ܹ衡 525ڡ 0:21:08 MAIL:ojamaru@amber.plala.or.jp 27471
äѥȥ
>DrK

ŷͤ㤦
525ڡ 0:25:10 27472
ERĹ
#27466ƱˡǤm(__)m
525ڡ 0:52:56 27473
Υܥ
ޥ뤵󡢤ޤޤͥڤ줷Ƥޤ͡
525ڡ 2:22:24 27474
ϥ饮㡼ƥ
Ϥ褦ޤ

MathematicaǥabȤ

Total[IntegerDigits[a*b]]

ȤФޤؿΤǤ
̶彣 525ڡ 7:14:13 HomePage:ĩ27475
⡼ޥ
Ʊǡ(^^)
525ڡ 7:34:02 MAIL:kennji72001@yahoo.co.jp 27476
uchinyan
ϤϤ褦ޤλ֤ʤΤǡˤäơŻǤ ^^/
66..(30)..66 * 66..(29)..68
= 4 * 33..(30)..33 * 33..(29)..34
= 4 * 33..(29)..34 * 3 * 11..(30)..11
= 4 * 100..(29)..02 * 11..(30)..11
= 4 * 11..(30)..1122..(30)..22
= 44..(30)..4488..(30)..88
ʤΤǡϡ
(4 + 8) * 30 = 360
ͥνȡ 525ڡ 8:19:01 MAIL:uchi@sco.bekkoame.ne.jp 27477
uchinyan
Ǽɤߤޤ׻ʤΤǡޤ򤽤äϤʤǤͤ
ξϡ66..66 顢2/3 = 0.666..., 1/3 = 0.333..., 3 * 1/3 = 1 = 0.999... ʤϢۤ졢
66...66 = 2 * 33...33, 66...68 = 2 * 33...34, 33...33 * 3 = 99...99, 33...34 * 3 = 100...02 㤡ʤ
Ȥǡ#27477̤Ǥ ^^;
ͥνȡ 525ڡ 9:13:08 MAIL:uchi@sco.bekkoame.ne.jp 27478
Υʼ
Ĥ⻻ϤǤʤΤ˺ϣʬǤǤޤ
äѥȥ뤵ƱǤޤ
525ڡ 10:28:40 MAIL:nagamikoko1000@gmail.com 27479
ޤ륱
פ֤Υ饤ʡ
ruby -e 'p eval((("6"*30).to_i * (("6"*30).to_i+2)).to_s.split("").join("+"))'
525ڡ 11:42:26 MAIL:take4310@mobile.email.ne.jp HomePage:ޤ륱27480
⡼ޥ
ʤߤˡ
(66...66)^2 ξϡ66...66*2=133...32 ʤΤǡ1+3*29+2=3*30=90 顢360-90=270 ʬޤ͡
525ڡ 13:00:06 MAIL:kennji72001@yahoo.co.jp 27481
粬ҹ
ߣᣴ
ߣᣴ
ߣᣴ
ηΣȣοƱͽۡʡ
äơʣ+ˡߣᣳ
Ϥʴǹͤޤʡ¾β⻲ͤˤƤ餤ޤ
525ڡ 13:52:26 MAIL:toshi009@land.hokuriku.ne.jp 27482
ryuta422
6x8=4812 66x68=448824 12x30=360

塡 塡
525ڡ 18:59:33 27483
ryuta422
䤸뤷ΰ֤ȤְäƤȤޤ
ޤ^^;
525ڡ 19:00:45 27484
tapu
פ֤˲򤱤ޤ줷Ǥ
525ڡ 20:09:48 27485
ȥҥ
ʤˤ줤ʿˤʤȤϻפ⤷ޤǤ
̾Ų 525ڡ 23:45:32 27486
uchinyan
()ϴñ褦ʤΤǡ꤫פĤϤɤǤ礦
ˤ顢ͤƤߤƤ
()äݤ(^^;

ˤ뼫򤫤顢ģΤ褦ˣ¤ּˤʤ뤳ȤǤ礦
ˤϡΤ褦ʼΣθĿƤ
ʣˤϡΤΰĤˤĤƵФΤȤޤ
ϡʤˤϣˤϤΥΣθĿȤޤ

ʤ񤷤⤷ޤ
ǽˡդǹͤƤߤȡ꤬꤬ĤȻפޤ

ʤ¤ϥҥȤǼĤν񤭹ߤˤޤ ^^;
ͥνȡ 526ʶ 13:39:04 MAIL:uchi@sco.bekkoame.ne.jp 27487
⡼ޥ
#27487
19^2=361<503<23^2=529 ǡ3~19 ޤǤǿdzʤΤǿ
10^5021 (mod 503)եޡξꡣ
顢9θĿϡ502ġ

˿漼ˡ(^^)

1/503=0.ab n ˤη֤Ȥȡ1/503=ab/99 (n ˤǤΤ
ȤС1/13=0.076923 η֤ʤΤǡ1/13=76923/999999
13*76923=999999 69
եޡξϡ10^12-1 13 dzڤ뤳Ȥ뤫顢129
ʾ夫顢Ǿ9η¸ߤȤϡp-1 ηˤ뤳Ȥϸ롣

1/503 ν۴ľϷ׻ѤǤޤ(^^;
äơ502=2*251 ʤΤǡȤƤ⡢251 ν۴ľʤǤ
526ʶ 18:11:19 MAIL:kennji72001@yahoo.co.jp 27488
uchinyan
#27488
ͭ̾ǤǤϡ¾οͤʬʤ⤷ʤʡ

ݥȤϡ#27478
ξϡ66..66 顢2/3 = 0.666..., 1/3 = 0.333..., 3 * 1/3 = 1 = 0.999... ʤϢۤ졢
դǤ1 = 3 * 1/3 = 3 * 0.333 = 0.999 ʤΤǡ
1/503 = 0.999 503 Ȥʤäơ1/503 γ껻ޤ
503 ǿʤΤǡ1/503 ۴ʬΤʤ۴ľȤʤޤ
۴ľν۴ʬΰĹϡ̤ˡ10^x 1 (mod p) x ˤʤޤ
ϡºݤ˳껻򤷤ƤߤʬȻפޤ
p ǿȤȡեޡξꡢɬ10^(p-1) 1 (mod p) Τǡ
ʤȤ⡢1/p ν۴ʬ p-1 ˤĹޤ
äơp-1 Ǥ⡢֤ޤ
ۤɤ 1/503 = 0.999 503 ȡդϡ99999 η뤳Ȥˤʤޤ
۴ʬȤ˰ڤˤʤΤǡ礦ɡ۴ʬĹڤӤܿ 9 θĿ 99999 ڤ뤳Ȥˤʤޤ
Ūˤϡ۴ʬĹ n Ȥơ
1/p = 0.abcdxyzabcdexyzabcdxyz
10^n * 1/p = abcdxyz.abcdexyzabcdxyz
(10^n - 1) * 1/p = abcdxyz
999999 = p * abcdxyz, 999999 9 n ¤Ǥ
ȤȤǤ͡
ξϡp = 503 ʤΤǡϡ502 ǤˤóݤƤߤޤ (^^;

ʤ503 ξ 502 Ǿ̤ˡp ξ p-1 ǾȤΤϡ
503 ξˤϡΤ 502 ǾˤʤäƤΤǤ̤ˤϡޤñǤϤʤǤ
ˡ¾ΥȤȯŸǸƤȤޤϡ10 p θϺȤΤˤʤäƤʤȥǡ
ȽϡޤưפǤϤʤ褦ǤʤȤ⡢ˤʬޤ
㤨С7 6 ǾǤ11 mod 11 10 -1, 10^2 1 Ȥʤꡢ۴ʬĹ 2 ˤʤޤ
⤷ɤʤȽˤĤƤ¸ΤǤ顢ꤤ
ͥνȡ 526ʶ 18:11:18 MAIL:uchi@sco.bekkoame.ne.jp 27489
uchinyan
#27488
1/503 ν۴ľϷ׻ѤǤޤ(^^;
ͤޤǤˡ
1/503 =
0.
0019880715705765407554671968190854870775347912524850894632206759443339960238568588469184890656063618
2902584493041749502982107355864811133200795228628230616302186878727634194831013916500994035785288270
3777335984095427435387673956262425447316103379721669980119284294234592445328031809145129224652087475
1491053677932405566600397614314115308151093439363817097415506958250497017892644135188866799204771371
7693836978131212723658051689860834990059642147117296222664015904572564612326043737574552683896620278
33
0019880715705765407554671968190854870775347912524850894632206759443339960238568588469184890656063618
2902584493041749502982107355864811133200795228628230616302186878727634194831013916500994035785288270
3777335984095427435387673956262425447316103379721669980119284294234592445328031809145129224652087475
1491053677932405566600397614314115308151093439363817097415506958250497017892644135188866799204771371
7693836978131212723658051689860834990059642147117296222664015904572564612326043737574552683896620278
33

Τˡ 502 Ƿ֤ޤ
ͥνȡ 526ʶ 18:19:39 MAIL:uchi@sco.bekkoame.ne.jp 27490
̴
ȿʤ(4/9)1060+1030-2)Фޤ
ǡ(4/9)*(99999(60)+99999(30))
44444(60)+44444(30)Ȥʤޤ
527ڡ 0:12:48 27491
̴
ȡ5/2518:43̾񤤤ơ֣פäƤϤʤΤ˥󥭥󥰤ˤʤǤʼ¤Ϥä˥ѥǤʤǤ衣
527ڡ 0:22:46 MAIL:max_and_yume_forever@yahoo.co.jp 27492
⡼ޥ
#27491
666=6(10^29+10^28++10+1)=6*1/9*(10^30-1)
668=6*1/9*(10^30-1)+2
666*668=4/9*(10^60+10^30-2) Ǥ
Ǥ͡

#27488
۴ľξ硢;ǤȤμ˳ϳʲǤ⤽οϴޤޤʤΤǡ
ȤС 503 ʤ顢ϡ1502 иʤϤǡĤޤϡ¿Ȥ⡢502 ޤǤˤϽ۴ĤϤȤȤϸޤ͡;Ǥ͡

ϺϿܤǤϴääݤ褦Ǥ錄⤤ޤԥƤޤ󡣡(^^;

527ڡ 10:47:59 MAIL:kennji72001@yahoo.co.jp 27493
uchinyan
#27491, #27493
ɤƱȤʤΤǤˤ۴ľȤСФʤƤ⡢ŪˡƱȤǤޤ
6/9 = 0.666 ʤΤǡ
666(30)666 = 666(30)666.666 - 0.666 = 0.666 * 10^30 - 0.666 = 6/9 * 10^30 - 6/9 = 6/9 * (10^30 - 1)
ϡƱǤ

٤ƥ˥åŪʹפϤ褦Ǥ
#27467, #27468, #27477 ʳˤ⡢äǤ
666(30)666 * 666(29)668
= 2 * 3 * 111(30)111 * 4 * 166(28)667
= 888(30)888 * 500(28)001
= 444(30)444888(30)888
ʤƤΤ⤢ޤ͡
֤ɤפȤ⤢ޤ(^^;
ͥνȡ 527ڡ 11:38:10 MAIL:uchi@sco.bekkoame.ne.jp 27494

ĩ路ΤǤߤʤϤɤ줰餤δĩʤäƤΤǤ

ޤǤФϡޤˤʤäƤޤäơ
ʤߤˤܤϡФǤ
527ڡ 21:53:59 27495
⡼ޥ
#27495
ء
錄Ͽǯ饢ʸƸĤޤˤ褦ˤʤޤǤϤˤϤޤäƤޤ(^^)
11ФȤФޤ5Ǥ͡ΰֲλҡ3ˤPCäƤ뤱ɡͧãȤΥ᡼ФäꤷƤޤ衣ϥ󥿡ͥåȤȤǤǤĶäƤޤ͡
528 8:14:27 MAIL:kennji72001@yahoo.co.jp 27496
⡼ޥ
פ֤ͧ

3򼡡ȤƼιब
1,3,9,27,81,243,729,
롣ο󤫤餳Ȥʤ뤤ĤοȤä­
Ǥ򾮤¤٤ƽ
1,3,4,9,10,12,13,
ͤȤο100ܤοϲ
528 13:34:01 MAIL:kennji72001@yahoo.co.jp 27497
uchinyan
#27497
׻㤤ʤС
981
ˡܻˡ
ͭ̾Ǥϡ㤤ʡ
ͥνȡ 528 14:07:35 MAIL:uchi@sco.bekkoame.ne.jp 27498

6666684448886666666844448888
66666666684444488888ǽϤǤ

528 14:17:44 27499
⡼ޥ
#27498
uchinyan

äͭ̾ʤǤ줤ǤΤ͡
ʤߤ˲С
Ϳ줿3ˡɽ
1101001000
ˤäƽ3ˡ
1101110010111011110001001
2ˡȤߤʤȼ1ʤ
֤ˤʤǤ롣
2ʿ100=1100100
äƵ3^6+3^5+3^2=981

528 14:22:24 MAIL:kennji72001@yahoo.co.jp 27500
uchinyan
#27500
ʤȤ⡢ΥȤǸϲ򤤤褦ʵƤޤ
Ǥ⡢Τʤ⤤ޤͤäȡ
٤顢ϡۤäƤޤ (^^;
ͥνȡ 528 15:00:04 MAIL:uchi@sco.bekkoame.ne.jp 27501
uchinyan
ȤǼĤǸǤϤɤǤ礦
ˤʤС

ΰ̤ǡΣƬˤäƤȡΣܤȤʤǾμηƤ
ɽˡ10ˡǤ
;͵СǾμΡǾǤϤʤϤɤʤΤͤƤߤƤ

ʤޤǤηǼĤȾطޤ
ͥνȡ 529ʷ 14:09:00 MAIL:uchi@sco.bekkoame.ne.jp 27502
⡼ޥ
#27502
>ΰ̤ǡΣƬˤäƤȡΣܤȤʤǾμηƤ
ɽˡ10ˡǤ
;͵СǾμΡǾǤϤʤϤɤʤΤͤƤߤƤ

׻ơ
210526315789473684 18塣
ǾǤʤϡη֤顢18ܿ塣

ȤοnȤ롣Ȥο10*x+4 Ȥȡx n-1 塣
4*10^(n-1)+x=2(10*x+4) 顢
19x=4*10^(n-1)-8
10^18-1=(10^9-1)(10^9+1)0 (mode 19)
Ĥޤꡢ19x ϡ18夫9夫3夬롣
1/19=x/999 ʤΤǡ׻ȡ
1/19=0.052631578947368421 ν۴ľ
x ϡ17 ʤΤǡȤοη18 Ǿʬ롣

x Ⱥǽ᤿ 21052631578947368(4) Ȱ㤦ʤ
529ʷ 17:58:54 MAIL:kennji72001@yahoo.co.jp 27503
uchinyan
#27503
⡼ޥ󤵤󡢲꤬ȤޤǤ ^^/
Ȥʤɡ

׻ơ
210526315789473684 18塣
η׻򾯤ܤ񤤤Ƥȡ
ϡ4Ƚ񤱤ޤ
4 ƬˤäƤȡ4 ǡ줬 2 * 4 = 8 Τǡ4 = 8 Ȥʤꡢ
Τ 28 Ƚ񤱤뤳ȤʬޤĤޤꡢ 284 ǤƱͤˤơ
284 -> 428 = 468 -> 2684 -> 4268 = 4368 -> 23684 -> 42368 = 47368 ->
273684 -> 427368 = 447368 -> 2473684 -> 4247368 = 4947368 -> 29473684 -> 42947368 = 48947368 ->
289473684 -> 428947368 = 478947368 -> 2789473684 -> 4278947368 = 4578947368 ->
25789473684 -> 42578947368 = 41578947368 -> 215789473684 -> 421578947368 = 431578947368 ->
2315789473684 -> 4231578947368 = 4631578947368 -> 26315789473684 -> 42631578947368 = 42631578947368 ->
226315789473684 -> 422631578947368 = 452631578947368 ->
2526315789473684 -> 4252631578947368 = 4052631578947368 ->
20526315789473684 -> 42052631578947368 = 41052631578947368 ->
210526315789473684 -> 421052631578947368 = 421052631578947368 -> Ȥˤʤ OK
ǡɡ210526315789473684 Ǿοǡ18ˤʤޤ

ơθεϡطʤọ̃ǽפǤ
ȤοnȤ롣Ȥο10*x+4 Ȥȡx n-1 塣
4*10^(n-1)+x=2(10*x+4) 顢
ɡx = (4 * 10^(n-1) - 8)/19 Ǥ͡
䤷x Ⱥǽ᤿ 21052631578947368(4) Ȱ㤦ʤ
Τϡx ǤϤʤ10 * x + 4 ǡ⡢x = (4 * 10^(n-1) - 8)/19 ʤΤǡ
ΤȤʤ顢1/19 ν۴ʬȤϰۤʤäƤޤˤĤƤϡDz⤷ޤ
ʳϽ񤫤Ƥ̤Ǥx ʤΤǡ
4 * 10^(n-1) - 8 0 (mod 19)
10^(n-1) 2 (mod 19)
Ƥ̤ꡢ10^k 1 (mod 19) κǾ k 18 ʤΤǡ
10^18 1 20 (mod 19)
10^17 2 (mod 19)
ȤʤꡢǾ n ϡn - 1 = 17 n = 18 ǡǾοη 18 ʤΤϡ
1/19 ξ10̤餤ޤǤη׻ʬޤ
äȤ⡢ǾοΤϡĥäơ۴ĤޤǤʤȥǤ
ʤ̲ϡ10^18 1 (mod 19) դơ
10 * (4 * 10^(n-1) - 8)/19 + 4, n n 0 (mod 18) Ȥʤ
Ȥ񤱤ޤ

ޤǤС۴ľȿط뤳ȤʬȻפޤϡ
1/19 = 0.052631578947368421052631578947368421
Ǥδط
2/19 = 20/10 * 1/19 = (1/10 + 19/10) * 1/19 = 1/10 * 1/19 + 1/10
4/19 = 2 * 2/19 = 1/10 * 2/19 + 2/10
Ȥȡ
1/19 = 000.052631578947368421052631578947368421
2/19 = 00.1052631578947368421052631578947368421
4/19 = 0.21052631578947368421052631578947368421
ʤΤǡ
10 * (4 * 10^(n-1) - 8)/19 + 4 = 10^n * 4/19 - 80/19 + 4 = 10^n * 4/19 - 4/19
顢ϡ4/19 ξʬǤä褦ˤФΤǡ
10^18 * 4/19 = 210526315789473684.21052631578947368421052631578947368421
00000 + 4/19 = 000000000000000000.21052631578947368421052631578947368421
ǡ00000 + פϡʸ·Τǰ̣Ϥޤ
ǡǾοϡ210526315789473684 ǡ 18 塢Ȥ뤳ȤǤޤ

ˤĤͷǤߤޤ۴ľϤ򤤤Ǥ ^^/
ͥνȡ 530ʲС 12:43:25 MAIL:uchi@sco.bekkoame.ne.jp 27504
⡼ޥ
#27504
1/19 = 0.052631578947368421 052631578947368421
2/19 = 0.105263157894736842 105263157894736842
3/19 = 0.157894736842105263 157894736842105263
4/19 = 0.210526315789473684 210526315789473684

ȡǽν۴ľ򤺤餷ΤˤʤäƤޤ͡
ʤʤΤͤƤޤ
껻򤷤ƤΤǡǿ19ξ硢118ޤɬФƤΤǡʽ۴ľη夬18夫ˡn/19 (1n18) ϡȤν۴ľ椫η׻򤹤뤳ȤƱ櫓餽ʤ櫓Ǥ͡
Ȥʤȡη夬4ʳǤ⤽ƬˤȤȤܤˤʤϡĤޤ͡
ʤߤ4ξϡ
736842105263157894
210526315789473684ΤΤ줫ʬꡢ
餫ԤʤԤǤȤ狼ޤ
Ԥϡ4Ƭ˻äƤȤȤ9ܤοˤʤޤ͡
530ʲС 13:50:23 MAIL:kennji72001@yahoo.co.jp 27505
uchinyan
#27505
껻򤷤ƤΤǡǿ19ξ硢118ޤɬФƤΤǡʽ۴ľη夬18夫ˡ
n/19 (1n18) ϡȤν۴ľ椫η׻򤹤뤳ȤƱ櫓餽ʤ櫓Ǥ͡
ϤȤǤ

736842105263157894
210526315789473684ΤΤ줫ʬꡢ
餫ԤʤԤǤȤ狼ޤ
736842105263157894 ----> 473684210526315789
736842105263157894 * 2 = 1473684210526315788
夬äǤΤǡŦ̤ꡢǰʤ顢ԤϥǤ͡
Ԥϡ4Ƭ˻äƤȤȤ9ܤοˤʤޤ͡
736842105263157894 ----> 473684210526315789
736842105263157894 * 9 = 6631578947368421046
ʤʤΤǤ褯ʬʤ
ͥνȡ 530ʲС 16:58:22 MAIL:uchi@sco.bekkoame.ne.jp 27507
⡼ޥ
#27507
>Ԥϡ4Ƭ˻äƤȤȤ9ܤοˤʤޤ͡
736842105263157894 ----> 473684210526315789
736842105263157894 * 9 = 6631578947368421046
ʤʤΤǤ褯ʬʤ

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ʲɤǤ⤤ȤǤ
ä;꤬1,10,5,12,6,3,11,15,17,18,9,14,7,13,16,8,4,2,
19ܤ181+18=19
109ܤ910+9=19




1/1911/19ϲ夬1Ʊ餺餷Τˤʤ롣
2/1912/19



8/19 18/19
9/19 ȡ
10/19 ȡ

9*2n Τ¤ֽ۴ľϤߤǿǤϤʤǤ͡
ޤοǤ¤٤Ȥɬ1Ĥ1/n ˤǤäƤȤǤͤ
18,36,54,72,90,108,126,144,162,180,198,216,234,
19,37,(55),73,(91),109,127,(145),163,181,199,217,(235),
530ʲС 22:45:29 MAIL:kennji72001@yahoo.co.jp 27508

奪ճŤǤ
531ʿ 13:54:04 MAIL:ryoiti@sansu.org 27509
ޥ
#27509
ϥ7/9ˤͽꤷƤޤˤʤä饢ʥ󥹤褦ȤפäƤΤǤɤäΤޤ(^^;
PowerBook G4 531ʿ 18:51:04 MAIL:masaru-y@sansu.org HomePage:27510