ta có : \(B=2004+2004^2+2004^3+...+2004^{10}\)

\(B=\left(2004+2004^2\right)+\left(2004^3+2004^4\right)+...+\left(2004^9+2004^{10}\right)\)

\(B=2004.\left(1+2004\right)+2004^3\left(1+2004\right)+...+2004^9\left(1+2004\right)\)

\(B=2004.2005+2004^3.2005+...+2004^9.2005\)

\(B=2005.\left(2004+2004^3+...+2004^9\right)⋮2005\)

\(\Rightarrow2005.\left(2004+2004^3+2004^9\right)\) chia hết cho \(2005\)

\(\Leftrightarrow B=2004+2004^2+2004^3+...+2004^{10}\) chia hết cho \(2005\) (đpcm)

(đpcm)

tầm như làm dạng này zùi

2005 n ≡1(mod167) 189 7 n ≡ 6 0 n ( m od 167 )

1897 n ≡60 n (mod167) 16 8 n ≡ 1 ( mo d 167 ) 168 n ≡1(mod167) ⇒A≡1+60 n −60 n −1≡0(mod167) ⇒A⋮167 Tương tự ta có: A ⋮ 4 A ⋮ 3 ⇒ A ⋮ 2004

C = 2004 + 2004²+2004³+2004⁴+...+2004¹⁰

C = (2004 +2004²)+(2004³+2004⁴)+...+(2004⁹+2004¹⁰)

C = 2004(1+2004) + 2004³ .(1+2004)+...+ 2004⁹. (1+2004)

C = 2004 .2005 + 2004³ .2005+....+2004⁹.2005

C = 2005.(2004+2004³ + ...+2004⁹)

Vì 2005 chia hết cho 2005 => 2005.(2004+2004³ + ...+2004⁹) chia hết cho 2005 => C chia hết cho 2005(ĐPCM)

Ta có : 

\(C=2004+2004^2+2004^3+...+2004^9+2004^{10}\)

\(=\left(2004+2004^2\right)+\left(2004^3+2004^4\right)+...+\left(2004^9+2004^{10}\right)\)

\(=2004\left(1+2004\right)+2004^3\left(1+2004\right)+...+2004^9\left(1+2004\right)\)

\(=2004.2005+2004^3.2005+...+2004^9.2005\)

\(=2005\left(2004+2004^3+...+2004^9\right)⋮2005\left(đpcm\right)\)

Ngô Thị Diệu Linh

C = 2004 + 20042+20043+20044+...+200410
C = (2004 +20042)+(20043+20044)+...+(20049+200410)

C = 2004(1+2004) + 20043 .(1+2004)+...+ 20049. (1+2004)

C = 2004 .2005 + 20043 .2005+....+20049.2005

C = 2005.(2004+20043 + ...+20049)

Vì 2005 chia hết cho 2005 => 2005.(2004+20043 + ...+20049) chia hết cho 2005 => C chia hết cho 2005(ĐPCM)

Ta có: 1.2.3.4...2004 = 1.2.3.4.5...401...2004 = [5.401].1.2.3.4.6....2004 = 2005.1.2.3....2004 chia hết cho 2005

=> Khi nhân với 1 + 1/2 + ... + 1/2004 cũng chia hết cho 2005

AI THẤY ĐÚNG NHỚ ỦNG HỘ

Ta có: \(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2004}\)

\(=\left(1+\frac{1}{2004}\right)+\left(\frac{1}{2}+\frac{1}{2003}\right)+\left(\frac{1}{3}+\frac{1}{2002}\right)+...+\left(\frac{1}{1002}+\frac{1}{1003}\right)\)

\(=\frac{2005}{1.2004}+\frac{2005}{2.2003}+\frac{2005}{3.2002}+...+\frac{2005}{1002.1003}\)

\(=2005\left(\frac{1}{1.2004}+\frac{1}{2.2003}+\frac{1}{3.2002}+....+\frac{1}{1002.1003}\right)\)

\(\Rightarrow A=1.2.3.....2004.\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{2004}\right)\)\(=1.2.3.....2004.2005\left(\frac{1}{1.2004}+\frac{1}{2.2003}+....+\frac{1}{1002.1003}\right)\)chia hết cho 2005 (đpcm)

`43^2004 + 43^2005 = 43^2004 (1 + 43) = 43^2004 . 44`

`=43^2004 . 4.11 \vdots 11`

`=>` ĐPCM.

\(43^{2004}+43^{2005}=43^{2004}\left(43+1\right)=44.43^{2004}⋮11\) do \(44⋮11\)