\(2011\equiv1\left(mod2010\right)\Rightarrow2011^{2009}\equiv1\left(mod2010\right)\)

\(2009\equiv-1\left(mod2010\right)\Rightarrow2009^{2011}\equiv-1\left(mod2010\right)\)

\(\Rightarrow2009^{2011}+2011^{2009}\equiv0\left(mod2010\right)\Rightarrow2009^{2011}+2011^{2009}⋮2010\)

mod là sao

Nó có chia hết à ??? 

Ta thấy 2009 chia 2010 dư  -1 

=> 2009 ^ 2008 chia 2010 dư 1 (1)

Lại có  2011 chia 2010 dư 1

=> 2011^2010 chia 2020 dư 1 (2)

Từ (1)(2) => 2009^2008-2011^2020 chia 2010 dư 2 (sai )

2009^2008+2011^2010 chia hết cho 2010 2009^2008+2011^2010

=2009^2008+2011^2010

=2009^2008+2011^2010+1-1

=(2009^2008+ 1) + (2011^2010– 1)

= (2009 + 1)(2009^2007- …) + (2011 – 1)(2011^2009 + …)

= 2010(2009^2008 - …) + 2010(2011^2009+ …) chia hết cho 2010  

\(2009^{2011}+2011^{2009}=\left(2009^{2011}+1\right)+\left(2011^{2009}-1\right)\)

Ta có: \(a^n+b^n⋮\left(a+b\right)\) với n là số lẻ.

\(a^n-b^n⋮\left(a-b\right)\forall n\inℕ^∗\)

Nên \(2009^{2011}+1⋮\left(2009+1\right),2011^{2009}-1⋮\left(2011-1\right)\)

Vậy \(2009^{2011}+1+2011^{2009}-1⋮2010\Rightarrow2009^{2011}+2011^{2009}⋮2010\)

Tại sao aⁿ+bⁿ chia hết a+b

From: exoplanet

To: Nguyễn Ngọc Phương Thảo

\(2009^{2008}+2011^{2010}=\left(2009^{2008}+1\right)+\left(2011^{2010}-1\right)\)

\(=\left(2009+1\right)\left(2009^{2007}+a\right)+\left(2011-1\right)\left(2011^{2009}-b\right)\)

\(2009^{2011}+1+2011^{2009}-1=\)   (2009+1)(2009²⁰¹⁰- 2009²⁰⁰⁹ +...- 2009+ 1)+(2011-1)(2011²⁰⁰⁸+2011²⁰⁰⁷+...+ 1) =

2010.A + 2010.B chia hết cho 2010