Ta có: 31+32+33+34+35+...+32012
=(3^1+3^2+3^3+3^4+3^5)+...+(3^2008+3^2009+^3^2010+3^2011+3^2012)
=(3*1+3*3+3*3^2+3*3^3+3*3^4)+...+(3^2008*1+3^2008*3+3^2008*3^2+3^2008*3^3+3^2008*3^4)
=3*(1+3+3^2+3^3+3^4)+....+3^2008*(1+3+3^2+3^3+3^4)
=3*121+...+3^2008*121
=(3+3^6+...+3^2008)*121
Vì 121 chia 120 dư 1
Nên 31+32+33+34+35+...+32012 chia hết cho 120
*là nhân nha bạn
Đặt S=\(3\)\(+\)\(3^2\)\(+\)\(3^3\)\(+\)...............\(+\)\(3^{2012}\)
\(\Rightarrow\)S=[\(3\)\(+\)\(3^2\)\(+\)\(3^3\)\(+\)]\(+\)........................\(+\)[\(3^{2009}\)\(+\)\(3^{2010}\)\(+\)\(3^{2011}\)\(+\)\(3^{2012}\)]
\(\Rightarrow\)S=120\(+\).......................\(+\)\(3^{2008}\)[\(3\)\(+\)\(3^2\)\(+\)\(3^3\)\(+\)\(3^4\)]
\(\Rightarrow\)S=120\(+\).......................\(+\)\(3^{2008}\)\(+\)120
\(\Rightarrow\)S=120[1\(+\)................\(+\)\(3^{2008}\)]
VÌ 120\(⋮\)120 \(\Rightarrow\)S\(⋮\)120
A=3+32+33+34+.......+3100
⇒A=(3+32+33+34)+.......+(397+398+399+3100)
⇒A=3.(1+3+32+33)+........+397.(1+3+32+33)
⇒A=3.40+.........+397.40
⇒A=40.(3+.......+397)
⇒A⋮40( 1 )
Vì A là tổng của các bậc lũy thừa của 3 nên A⋮3 ( 2 )
Từ ( 1 ) và ( 2 ) suy ra : A⋮ 40.3
⇒A⋮120
Vậy A⋮120( ĐPCM )