Question

1.aflati ultima cifra a nr: a=2001^2007+2002^2002+2003^2003+2004^2004 = b) 2+2^2+2^3+2^4+...+2^17 = 2. Aflati ultimele trei cifre ale nr: m=7^77-7^76+7^75-7^74= ( ^semnul asta inseamna la puterea)

Answer 1

a=2001^2007+2002^2002+2003^2003+2004^2004
Uc(2001^2007)=Uc(1^2007)=1
Uc(2002^2002)=Uc(2^2002)=Uc[2^(4*500+2)]=Uc(2^2)=4
Uc(2003^2003)=Uc(3^2003)=Uc[3^(4*500+3)]=Uc(3^3)=7
Uc(2004^2004)=Uc(4^2004)=Uc[4^(2*1002)]=Uc(4^2)=6
1+4+7+6=18 ⇒Uc(a)=8

b= 2+2^2+2^3+2^4+...+2^17 
17:4=4*4 grupuri de(2,4,8,6)+1
Uc( 2+2^2+2^3+2^4)=2+4+8+16=0
Uc(b)=4*0+2^1= 2

2.m=7^77-7^76+7^75-7^74=7^74(7^3-7^2+7^1-7^0)=7^74(343-49+7-1)=
=7^74×300
Uc(7^74)=Uc(7^2)=9
9×300=2700
U₃(m)=700