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連続する5つの整数の積は120で割り切れることを証明せよ

予想外にコメントが…といっても4つだが、いただけたのでもう少し考えてみる。 タイトルに書いたように、5つの積は120、のようになったら爆発的なことになるので美しくない、 というご指摘を頂いた。 確かにそうだ。 実際にやってみた。

証明

任意の整数は、nを整数としたとき、 120n、120n+1、120n+2、120n+3、120n+4、120n+5、120n+6、120n+7、120n+8、120n+9、120n+10、 120n+11、120n+12、120n+13、120n+14、120n+15、120n+16、120n+17、120n+18、120n+19、120n+20、 120n+21、120n+22、120n+23、120n+24、120n+25、120n+26、120n+27、120n+28、120n+29、120n+30、 120n+31、120n+32、120n+33、120n+34、120n+35、120n+36、120n+37、120n+38、120n+39、120n+40、 120n+41、120n+42、120n+43、120n+44、120n+45、120n+46、120n+47、120n+48、120n+49、120n+50、 120n+51、120n+52、120n+53、120n+54、120n+55、120n+56、120n+57、120n+58、120n+59、120n+60、 120n+61、120n+62、120n+63、120n+64、120n+65、120n+66、120n+67、120n+68、120n+69、120n+70、 120n+71、120n+72、120n+73、120n+74、120n+75、120n+76、120n+77、120n+78、120n+79、120n+80、 120n+81、120n+82、120n+83、120n+84、120n+85、120n+86、120n+87、120n+88、120n+89、120n+90、 120n+91、120n+92、120n+93、120n+94、120n+95、120n+96、120n+97、120n+98、120n+99、120n+100、 120n+101、120n+102、120n+103、120n+104、120n+105、120n+106、120n+107、120n+108、120n+109、120n+110、 120n+111、120n+112、120n+113、120n+114、120n+115、120n+116、120n+117、120n+118、120n+119、 のいずれかで表現できる。

何かリピーターの人が半分減ったような気がするのだが、 まあそれはそうとして、 確かにこれでは採点する方もイヤだよな、という感じがしなくもない。 もうこの段階でやめておきたいような気もするが、 中途半端はよくないのである。

120n(120n+1)(120n+2)(120n+3)(120n+4) は、120で割り切れる。さらに、
(120n+1)(120n+2)(120n+3)(120n+4)(120n+5) = 120(120n+1)(60n+1)(40n+1)(30n+1)(24n+1)
(120n+2)(120n+3)(120n+4)(120n+5)(120n+6) = 120(60n+1)(40n+1)(30n+1)(24n+1)(120n+6)
(120n+3)(120n+4)(120n+5)(120n+6)(120n+7) = 120(40n+1)(30n+1)(24n+1)(60n+3)(120n+7)
(120n+4)(120n+5)(120n+6)(120n+7)(120n+8) = 120(30n+1)(24n+1)(20n+1)(120n+7)(120n+8)
(120n+5)(120n+6)(120n+7)(120n+8)(120n+9) = 120(24n+1)(20n+1)(120n+7)(30n+2)(120n+9)
(120n+6)(120n+7)(120n+8)(120n+9)(120n+10) = 120(20n+1)(120n+7)(30n+2)(120n+9)(24n+2)
(120n+7)(120n+8)(120n+9)(120n+10)(120n+11) = 120(120n+7)(15n+1)(40n+3)(24n+2)(120n+11)
(120n+8)(120n+9)(120n+10)(120n+11)(120n+12) = 120(15n+1)(40n+3)(24n+2)(120n+11)(120n+12)
(120n+9)(120n+10)(120n+11)(120n+12)(120n+13) = 120(40n+3)(12n+1)(120n+11)(30n+3)(120n+13)
(120n+10)(120n+11)(120n+12)(120n+13)(120n+14) = 120(12n+1)(120n+11)(10n+1)(120n+13)(120n+14)
(120n+11)(120n+12)(120n+13)(120n+14)(120n+15) = 120(120n+11)(10n+1)(120n+13)(60n+7)(24n+3)
(120n+12)(120n+13)(120n+14)(120n+15)(120n+16) = 120(10n+1)(120n+13)(60n+7)(24n+3)(120n+16)
(120n+13)(120n+14)(120n+15)(120n+16)(120n+17) = 120(120n+13)(60n+7)(8n+1)(30n+4)(120n+17)
(120n+14)(120n+15)(120n+16)(120n+17)(120n+18) = 120(60n+7)(8n+1)(30n+4)(120n+17)(120n+18)
(120n+15)(120n+16)(120n+17)(120n+18)(120n+19) = 120(8n+1)(15n+2)(120n+17)(120n+18)(120n+19)
(120n+16)(120n+17)(120n+18)(120n+19)(120n+20) = 120(15n+2)(120n+17)(40n+6)(120n+19)(24n+4)
(120n+17)(120n+18)(120n+19)(120n+20)(120n+21) = 120(120n+17)(20n+3)(120n+19)(6n+1)(120n+21)
(120n+18)(120n+19)(120n+20)(120n+21)(120n+22) = 120(20n+3)(120n+19)(6n+1)(120n+21)(120n+22)
(120n+19)(120n+20)(120n+21)(120n+22)(120n+23) = 120(120n+19)(6n+1)(40n+7)(60n+11)(120n+23)
(120n+20)(120n+21)(120n+22)(120n+23)(120n+24) = 120(6n+1)(40n+7)(60n+11)(120n+23)(120n+24)
(120n+21)(120n+22)(120n+23)(120n+24)(120n+25) = 120(40n+7)(60n+11)(120n+23)(30n+6)(24n+5)
(120n+22)(120n+23)(120n+24)(120n+25)(120n+26) = 120(60n+11)(120n+23)(10n+2)(24n+5)(120n+26)
(120n+23)(120n+24)(120n+25)(120n+26)(120n+27) = 120(120n+23)(5n+1)(24n+5)(120n+26)(120n+27)
(120n+24)(120n+25)(120n+26)(120n+27)(120n+28) = 120(5n+1)(24n+5)(120n+26)(120n+27)(120n+28)
(120n+25)(120n+26)(120n+27)(120n+28)(120n+29) = 120(24n+5)(60n+13)(40n+9)(30n+7)(120n+29)
(120n+26)(120n+27)(120n+28)(120n+29)(120n+30) = 120(60n+13)(40n+9)(30n+7)(120n+29)(24n+6)
(120n+27)(120n+28)(120n+29)(120n+30)(120n+31) = 120(40n+9)(30n+7)(120n+29)(12n+3)(120n+31)
(120n+28)(120n+29)(120n+30)(120n+31)(120n+32) = 120(30n+7)(120n+29)(4n+1)(120n+31)(120n+32)
(120n+29)(120n+30)(120n+31)(120n+32)(120n+33) = 120(120n+29)(4n+1)(120n+31)(30n+8)(120n+33)
(120n+30)(120n+31)(120n+32)(120n+33)(120n+34) = 120(4n+1)(120n+31)(30n+8)(120n+33)(120n+34)
(120n+31)(120n+32)(120n+33)(120n+34)(120n+35) = 120(120n+31)(15n+4)(40n+11)(120n+34)(24n+7)
(120n+32)(120n+33)(120n+34)(120n+35)(120n+36) = 120(15n+4)(40n+11)(120n+34)(24n+7)(120n+36)
(120n+33)(120n+34)(120n+35)(120n+36)(120n+37) = 120(40n+11)(60n+17)(24n+7)(30n+9)(120n+37)
(120n+34)(120n+35)(120n+36)(120n+37)(120n+38) = 120(60n+17)(24n+7)(10n+3)(120n+37)(120n+38)
(120n+35)(120n+36)(120n+37)(120n+38)(120n+39) = 120(24n+7)(10n+3)(120n+37)(60n+19)(120n+39)
(120n+36)(120n+37)(120n+38)(120n+39)(120n+40) = 120(10n+3)(120n+37)(60n+19)(120n+39)(24n+8)
(120n+37)(120n+38)(120n+39)(120n+40)(120n+41) = 120(120n+37)(60n+19)(40n+13)(6n+2)(120n+41)
(120n+38)(120n+39)(120n+40)(120n+41)(120n+42) = 120(60n+19)(40n+13)(6n+2)(120n+41)(120n+42)
(120n+39)(120n+40)(120n+41)(120n+42)(120n+43) = 120(40n+13)(3n+1)(120n+41)(120n+42)(120n+43)
(120n+40)(120n+41)(120n+42)(120n+43)(120n+44) = 120(3n+1)(120n+41)(40n+14)(120n+43)(120n+44)
(120n+41)(120n+42)(120n+43)(120n+44)(120n+45) = 120(120n+41)(20n+7)(120n+43)(30n+11)(24n+9)
(120n+42)(120n+43)(120n+44)(120n+45)(120n+46) = 120(20n+7)(120n+43)(30n+11)(24n+9)(120n+46)
(120n+43)(120n+44)(120n+45)(120n+46)(120n+47) = 120(120n+43)(30n+11)(8n+3)(60n+23)(120n+47)
(120n+44)(120n+45)(120n+46)(120n+47)(120n+48) = 120(30n+11)(8n+3)(60n+23)(120n+47)(120n+48)
(120n+45)(120n+46)(120n+47)(120n+48)(120n+49) = 120(8n+3)(60n+23)(120n+47)(30n+12)(120n+49)
(120n+46)(120n+47)(120n+48)(120n+49)(120n+50) = 120(60n+23)(120n+47)(10n+4)(120n+49)(24n+10)
(120n+47)(120n+48)(120n+49)(120n+50)(120n+51) = 120(120n+47)(5n+2)(120n+49)(24n+10)(120n+51)
(120n+48)(120n+49)(120n+50)(120n+51)(120n+52) = 120(5n+2)(120n+49)(24n+10)(120n+51)(120n+52)
(120n+49)(120n+50)(120n+51)(120n+52)(120n+53) = 120(120n+49)(12n+5)(40n+17)(30n+13)(120n+53)
(120n+50)(120n+51)(120n+52)(120n+53)(120n+54) = 120(12n+5)(40n+17)(30n+13)(120n+53)(120n+54)
(120n+51)(120n+52)(120n+53)(120n+54)(120n+55) = 120(40n+17)(30n+13)(120n+53)(60n+27)(24n+11)
(120n+52)(120n+53)(120n+54)(120n+55)(120n+56) = 120(30n+13)(120n+53)(20n+9)(24n+11)(120n+56)
(120n+53)(120n+54)(120n+55)(120n+56)(120n+57) = 120(120n+53)(20n+9)(24n+11)(30n+14)(120n+57)
(120n+54)(120n+55)(120n+56)(120n+57)(120n+58) = 120(20n+9)(24n+11)(30n+14)(120n+57)(120n+58)
(120n+55)(120n+56)(120n+57)(120n+58)(120n+59) = 120(24n+11)(15n+7)(40n+19)(120n+58)(120n+59)
(120n+56)(120n+57)(120n+58)(120n+59)(120n+60) = 120(15n+7)(40n+19)(120n+58)(120n+59)(24n+12)
(120n+57)(120n+58)(120n+59)(120n+60)(120n+61) = 120(40n+19)(60n+29)(120n+59)(6n+3)(120n+61)
(120n+58)(120n+59)(120n+60)(120n+61)(120n+62) = 120(60n+29)(120n+59)(2n+1)(120n+61)(120n+62)
(120n+59)(120n+60)(120n+61)(120n+62)(120n+63) = 120(120n+59)(2n+1)(120n+61)(60n+31)(120n+63)
(120n+60)(120n+61)(120n+62)(120n+63)(120n+64) = 120(2n+1)(120n+61)(60n+31)(120n+63)(120n+64)
(120n+61)(120n+62)(120n+63)(120n+64)(120n+65) = 120(120n+61)(60n+31)(40n+21)(30n+16)(24n+13)
(120n+62)(120n+63)(120n+64)(120n+65)(120n+66) = 120(60n+31)(40n+21)(30n+16)(24n+13)(120n+66)
(120n+63)(120n+64)(120n+65)(120n+66)(120n+67) = 120(40n+21)(15n+8)(24n+13)(120n+66)(120n+67)
(120n+64)(120n+65)(120n+66)(120n+67)(120n+68) = 120(15n+8)(24n+13)(40n+22)(120n+67)(120n+68)
(120n+65)(120n+66)(120n+67)(120n+68)(120n+69) = 120(24n+13)(20n+11)(120n+67)(30n+17)(120n+69)
(120n+66)(120n+67)(120n+68)(120n+69)(120n+70) = 120(20n+11)(120n+67)(30n+17)(120n+69)(24n+14)
(120n+67)(120n+68)(120n+69)(120n+70)(120n+71) = 120(120n+67)(30n+17)(40n+23)(12n+7)(120n+71)
(120n+68)(120n+69)(120n+70)(120n+71)(120n+72) = 120(30n+17)(40n+23)(12n+7)(120n+71)(120n+72)
(120n+69)(120n+70)(120n+71)(120n+72)(120n+73) = 120(40n+23)(12n+7)(120n+71)(30n+18)(120n+73)
(120n+70)(120n+71)(120n+72)(120n+73)(120n+74) = 120(12n+7)(120n+71)(10n+6)(120n+73)(120n+74)
(120n+71)(120n+72)(120n+73)(120n+74)(120n+75) = 120(120n+71)(5n+3)(120n+73)(120n+74)(24n+15)
(120n+72)(120n+73)(120n+74)(120n+75)(120n+76) = 120(5n+3)(120n+73)(120n+74)(24n+15)(120n+76)
(120n+73)(120n+74)(120n+75)(120n+76)(120n+77) = 120(120n+73)(60n+37)(8n+5)(30n+19)(120n+77)
(120n+74)(120n+75)(120n+76)(120n+77)(120n+78) = 120(60n+37)(8n+5)(30n+19)(120n+77)(120n+78)
(120n+75)(120n+76)(120n+77)(120n+78)(120n+79) = 120(8n+5)(30n+19)(120n+77)(60n+39)(120n+79)
(120n+76)(120n+77)(120n+78)(120n+79)(120n+80) = 120(30n+19)(120n+77)(20n+13)(120n+79)(24n+16)
(120n+77)(120n+78)(120n+79)(120n+80)(120n+81) = 120(120n+77)(20n+13)(120n+79)(6n+4)(120n+81)
(120n+78)(120n+79)(120n+80)(120n+81)(120n+82) = 120(20n+13)(120n+79)(6n+4)(120n+81)(120n+82)
(120n+79)(120n+80)(120n+81)(120n+82)(120n+83) = 120(120n+79)(3n+2)(40n+27)(120n+82)(120n+83)
(120n+80)(120n+81)(120n+82)(120n+83)(120n+84) = 120(3n+2)(40n+27)(120n+82)(120n+83)(120n+84)
(120n+81)(120n+82)(120n+83)(120n+84)(120n+85) = 120(40n+27)(60n+41)(120n+83)(30n+21)(24n+17)
(120n+82)(120n+83)(120n+84)(120n+85)(120n+86) = 120(60n+41)(120n+83)(10n+7)(24n+17)(120n+86)
(120n+83)(120n+84)(120n+85)(120n+86)(120n+87) = 120(120n+83)(10n+7)(24n+17)(60n+43)(120n+87)
(120n+84)(120n+85)(120n+86)(120n+87)(120n+88) = 120(10n+7)(24n+17)(60n+43)(120n+87)(120n+88)
(120n+85)(120n+86)(120n+87)(120n+88)(120n+89) = 120(24n+17)(60n+43)(40n+29)(30n+22)(120n+89)
(120n+86)(120n+87)(120n+88)(120n+89)(120n+90) = 120(60n+43)(40n+29)(30n+22)(120n+89)(24n+18)
(120n+87)(120n+88)(120n+89)(120n+90)(120n+91) = 120(40n+29)(15n+11)(120n+89)(24n+18)(120n+91)
(120n+88)(120n+89)(120n+90)(120n+91)(120n+92) = 120(15n+11)(120n+89)(8n+6)(120n+91)(120n+92)
(120n+89)(120n+90)(120n+91)(120n+92)(120n+93) = 120(120n+89)(4n+3)(120n+91)(30n+23)(120n+93)
(120n+90)(120n+91)(120n+92)(120n+93)(120n+94) = 120(4n+3)(120n+91)(30n+23)(120n+93)(120n+94)
(120n+91)(120n+92)(120n+93)(120n+94)(120n+95) = 120(120n+91)(30n+23)(40n+31)(60n+47)(24n+19)
(120n+92)(120n+93)(120n+94)(120n+95)(120n+96) = 120(30n+23)(40n+31)(60n+47)(24n+19)(120n+96)
(120n+93)(120n+94)(120n+95)(120n+96)(120n+97) = 120(40n+31)(60n+47)(24n+19)(30n+24)(120n+97)
(120n+94)(120n+95)(120n+96)(120n+97)(120n+98) = 120(60n+47)(24n+19)(10n+8)(120n+97)(120n+98)
(120n+95)(120n+96)(120n+97)(120n+98)(120n+99) = 120(24n+19)(5n+4)(120n+97)(120n+98)(120n+99)
(120n+96)(120n+97)(120n+98)(120n+99)(120n+100) = 120(5n+4)(120n+97)(120n+98)(120n+99)(24n+20)
(120n+97)(120n+98)(120n+99)(120n+100)(120n+101) = 120(120n+97)(60n+49)(40n+33)(6n+5)(120n+101)
(120n+98)(120n+99)(120n+100)(120n+101)(120n+102) = 120(60n+49)(40n+33)(6n+5)(120n+101)(120n+102)
(120n+99)(120n+100)(120n+101)(120n+102)(120n+103) = 120(40n+33)(6n+5)(120n+101)(60n+51)(120n+103)
(120n+100)(120n+101)(120n+102)(120n+103)(120n+104) = 120(6n+5)(120n+101)(20n+17)(120n+103)(120n+104)
(120n+101)(120n+102)(120n+103)(120n+104)(120n+105) = 120(120n+101)(20n+17)(120n+103)(30n+26)(24n+21)
(120n+102)(120n+103)(120n+104)(120n+105)(120n+106) = 120(20n+17)(120n+103)(30n+26)(24n+21)(120n+106)
(120n+103)(120n+104)(120n+105)(120n+106)(120n+107) = 120(120n+103)(15n+13)(8n+7)(120n+106)(120n+107)
(120n+104)(120n+105)(120n+106)(120n+107)(120n+108) = 120(15n+13)(8n+7)(120n+106)(120n+107)(120n+108)
(120n+105)(120n+106)(120n+107)(120n+108)(120n+109) = 120(8n+7)(60n+53)(120n+107)(30n+27)(120n+109)
(120n+106)(120n+107)(120n+108)(120n+109)(120n+110) = 120(60n+53)(120n+107)(10n+9)(120n+109)(24n+22)
(120n+107)(120n+108)(120n+109)(120n+110)(120n+111) = 120(120n+107)(10n+9)(120n+109)(12n+11)(120n+111)
(120n+108)(120n+109)(120n+110)(120n+111)(120n+112) = 120(10n+9)(120n+109)(12n+11)(120n+111)(120n+112)
(120n+109)(120n+110)(120n+111)(120n+112)(120n+113) = 120(120n+109)(12n+11)(40n+37)(30n+28)(120n+113)
(120n+110)(120n+111)(120n+112)(120n+113)(120n+114) = 120(12n+11)(40n+37)(30n+28)(120n+113)(120n+114)
(120n+111)(120n+112)(120n+113)(120n+114)(120n+115) = 120(40n+37)(15n+14)(120n+113)(120n+114)(24n+23)
(120n+112)(120n+113)(120n+114)(120n+115)(120n+116) = 120(15n+14)(120n+113)(40n+38)(24n+23)(120n+116)
(120n+113)(120n+114)(120n+115)(120n+116)(120n+117) = 120(120n+113)(20n+19)(24n+23)(30n+29)(120n+117)
(120n+114)(120n+115)(120n+116)(120n+117)(120n+118) = 120(20n+19)(24n+23)(30n+29)(120n+117)(120n+118)
(120n+115)(120n+116)(120n+117)(120n+118)(120n+119) = 120(24n+23)(30n+29)(40n+39)(60n+59)(120n+119)
(120n+116)(120n+117)(120n+118)(120n+119)(120n+120) = 120(30n+29)(40n+39)(60n+59)(120n+119)(24n+24)
(120n+117)(120n+118)(120n+119)(120n+120)(120n+121) = 120(40n+39)(60n+59)(120n+119)(6n+6)(120n+121)
(120n+118)(120n+119)(120n+120)(120n+121)(120n+122) = 120(60n+59)(120n+119)(2n+2)(120n+121)(120n+122)
(120n+119)(120n+120)(120n+121)(120n+122)(120n+123) = 120(120n+119)(n+1)(120n+121)(120n+122)(120n+123)

従って連続する5つの整数の積は常に120で割り切れる、

何か証明になっているような気がしない。 もちろんこんなの手で書いてられないからプログラムで作っている。 だから逆に間違いはないと思うのだが、 だったら証明用のプログラムを書いた方がお互い実りあるのでは、というような気がする。

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« 勉強する意欲が出ないのでモチベーションが欲しいという中高生の人達に | トップページ | 昨夜21:00~22:00頃東京の某所で大きな雷のような音がした »

コメント

むぅ。
どうせなら「連続する6つの整数の積は常に720で割り切れる」まで書いとけば良かった(違)(^^;

投稿: <セルダン> | 2009.04.01 19:14

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