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M=(2010+2010^2)+(2010^3+2010^4)+(2010^5+2010^6)+2010^7+1
=2010x2011+2010^3x2011+2010^5x2011+2010^7+1
=2011x(2010+2010^3+2010^5)+2010^7+1
mà 2010^6 đồng dư với 1 (mod 2011) nen 2010^6 x 2010 dong du voi 2010(mod 2011)
nên 2010^6 x 2010 +1 đồng dư với 2011 (mod 2011) nên 2010^7 +1 chia hết cho 2011 vậy m chia hết cho 2011
dễ ợt
s=2010(1+20100+2010^3(1+2010)+............+2010^2009(1+2010)
s=2010.2011+2010^3.2011+.........+2010^2009.2011
s=2011(2010+2010^3+.......+2010^2009) chia hết cho 2011
\(S=\left(2010+2010^2\right)+\left(2010^3+2010^4\right)+...+\left(2010^{2009}+2010^{2010}\right)\)
\(S=2010\left(2010+1\right)+2010^3\left(2010+1\right)+...+2010^{2009}\left(2010+1\right)\)
\(S=2011.\left(2010+2010^3+2010^5+...+2010^{2009}\right)\) chia hết cho 2011
A=2010^1+2010^2+2010^3+..........................................+2010^2010
vay suy ra co tat ca 2010 s hang vay ghep cap
A=2010(1+2010)+2010^3(1+2010)+..........................+2010^9(1+2010)
A=2010.2011+2010^3.2011+............................+2010^9.2011
A=2011(2010+........2010^9) chia het 2011
suy ra A chia het cho 2011
\(T=2010\left(1+2010\right)+2010^3\left(1+2010\right)+....+2010^{2009}\left(1+2010\right)\)
\(=2010.2011+...+2010^{2009}.2011\) chia hết cho 2011
=>đpcm
M= ( 1+20101)+(20102+20103)+(20104+20105)+(20106+20107)
M= 1.(2010+1) + 20122.(2010+1)+20104.(2010+1)+20106.(2010+1)
M= 2011.(1+20122+20104+20106)
Vậy M chia hết cho 2011