Epa! Acabei de reparar que errei oo copiar o enunciado. Era pra descobrir o ultimo algarismo nao nulo de 10! e nao de 1000!. Espero que esse pequeno engano nao tenha causado nenhum transtorno...
[]s, Claudio. > fyi... > > 1000! = > 2^994*3^498*5^249*7^164*11^98*13^81*17^61*19^54*23^44*29^35*31^33*37^27* > 41^24*43^23*47^21*53^18*59^16*61^16*67^14*71^14*73^13*79^12*83^12*89^11* > 97^10*101^9*103^9*107^9*109^9*113^8*127^7*131^7*137^7*139^7*149^6*151^6* > 157^6*163^6*167^5*173^5*179^5*181^5*191^5*193^5*197^5*199^5*211^4*223^4* > 227^4*229^4*233^4*239^4*241^4*251^3*257^3*263^3*269^3*271^3*277^3*281^3* > 283^3*293^3*307^3*311^3*313^3*317^3*331^3*337^2*347^2*349^2*353^2*359^2* > 367^2*373^2*379^2*383^2*389^2*397^2*401^2*409^2*419^2*421^2*431^2*433^2* > 439^2*443^2*449^2*457^2*461^2*463^2*467^2*479^2*487^2*491^2*499^2*503^1* > 509^1*521^1*523^1*541^1*547^1*557^1*563^1*569^1*571^1*577^1*587^1*593^1* > 599^1*601^1*607^1*613^1*617^1*619^1*631^1*641^1*643^1*647^1*653^1*659^1* > 661^1*673^1*677^1*683^1*691^1*701^1*709^1*719^1*727^1*733^1*739^1*743^1* > 751^1*757^1*761^1*769^1*773^1*787^1*797^1*809^1*811^1*821^1*823^1*827^1* > 829^1*839^1*853^1*857^1*859^1*863^1*877^1*881^1*883^1*887^1*907^1*911^1* > 919^1*929^1*937^1*941^1*947^1*953^1*967^1*971^1*977^1*983^1*991^1*997^1 > > dividindo por 10^249 e somando todos os expoentes de fatores terminados em > 3,7,9 e 1 temos: > > 2^745*3^772*7^388*9^206[*1^whocares] > > como > 2^(4k+1) = 6^k*2 = 6*2 = 2 (mod 10) -> 2^745 = 2 (mod 10) > 3^(4k)=1^k =1 (mod 10) -> 3^772 = 1 (mod 10) > 7^(4k)=1^k = 1 (mod 10) -> 7^388 = 1 (mod 10) > 9^(2k)= (-1)^(2k) = 1^2k = 1 (mod 10) -> 9^206 = 1 (mod 10) > > entao > 2^745*3^772*7^388*9^206 = 2*1*1*1 = 2 (mod 10) > > []s, > Campeao de Inutilidades > >> From: "Cl�udio \(Pr�tica\)" <[EMAIL PROTECTED]> >> Reply-To: [EMAIL PROTECTED] >> To: <[EMAIL PROTECTED]> >> Subject: [obm-l] Digitos de 1000! >> Date: Mon, 22 Mar 2004 12:08:54 -0300 >> >> HelpOi, pessoal: >> >> J� que o assunto � pot�ncias de primos que dividem n!, aqui vai um >> bonitinho: >> >> 1) (cl�ssico) Por quantos zeros termina a representa��o decimal de 1000! >> >> 2) (menos conhecido e mais dif�cil) Qual o �ltimo algarismo n�o nulo na >> representa��o decimal de 1000! >> >> 3) (generaliza��o) Qual o �ltimo algarismo n�o nulo de n!? Dica n�o muito >> �til: � sempre par se n > 1. >> >> >> []s, >> Claudio. >> > ========================================================================= Instru��es para entrar na lista, sair da lista e usar a lista em http://www.mat.puc-rio.br/~nicolau/olimp/obm-l.html =========================================================================
