đặt \(S=1+4+4^2+......+4^{1999}\)

\(\Rightarrow4S=4+4^2+4^3+....+4^{2000}\)

\(\Rightarrow4S-S=\left(4+4^2+4^3+....+4^{2000}\right)-\left(1+4+4^2+.....+4^{1999}\right)\)

\(\Rightarrow3S=4^{2000}-1\Rightarrow S=\frac{4^{2000}-1}{3}\)

Khi đó \(A=75.S=75.\frac{4^{2000}-1}{3}=\frac{75.\left(4^{2000}-1\right)}{3}=\frac{75}{3}.\left(4^{2000}-1\right)=25.\left(4^{2000}-1\right)=25.4^{2000}-25\)

Ta có: 4²⁰⁰⁰-1=(4⁴)⁵⁰⁰-1=(...6)-1=....5

=>25.4²⁰⁰⁰-25=25.(....5)-25=(...5)-25=....0 chia hết cho 100

Vậy ta có điều phải chứng minh
 

75 chia hết cho 25.

4²⁰⁰⁷ + ... + 4 + 1 chia 4 dư 1 hay không chia hết cho 4

=> 75(4²⁰⁰⁷ + ... + 4 + 1) không chia hết cho 100.

75 chia hết cho 25.

4²⁰⁰⁷ + ... + 4 + 1 chia 4 dư 1 hay không chia hết cho 4

=> 75(4²⁰⁰⁷ + ... + 4 + 1) không chia hết cho 100.

Đặt \(B=1+4+4^2+...+4^{1998}+4^{1999}\)

\(\Rightarrow4B=4+4^2+4^3+...+4^{1999}+4^{2000}\)

\(\Rightarrow4B-B=\left(4+4^2+4^3+...+4^{2000}\right)-\left(1+4+4^2+...+4^{1999}\right)\)

\(\Rightarrow3B=4^{2000}-1\)

\(\Rightarrow B=\dfrac{4^{2000}-1}{3}\)

Khi đó ta có:

\(A=75.B=75.\dfrac{4^{2000}-1}{3}=\dfrac{75.\left(4^{2000}-1\right)}{3}=\dfrac{75}{3}.\left(4^{2000}-1\right)=25.\left(4^{2000}-1\right)=25.4^{2000}-25\)

Ta có: \(4^{2000}-1=\left(4^4\right)^{500}-1=\left(...6\right)-1=...5\)

\(\Rightarrow25.4^{2000}-25=25.\left(...5\right)-25=\left(...5\right)-25=...0⋮100\left(đpcm\right)\)

Ta có:

\(A=75.\left(4^{1999}+4^{1998}+...+4^2+4+1\right)+25\)

\(A=25.3.\left(4^{1999}+4^{1998}+...+4^2+4+1\right)+25\) \(A=25.\left(4-1\right).\left(4^{1999}+4^{1998}+...+4^2+4+1\right)+25\)

\(A=25.\left(4^{2000}+4^{1999}+...+4^3+4^2+4-4^{1999}-4^{1998}-...-4^2-4-1\right)+25\)\(A=25.\left(4^{2000}-1\right)+25\)

\(A=25.\left(4^{2000}-1+1\right)\)

\(A=25.4^{2000}=25.4.4^{1999}=100.4^{1999}\)Vây:A là số chia hết cho 100

1) A = 1997¹⁹⁹⁹ - 1997¹⁹⁹⁸

=> A = 1997¹⁹⁹⁸.(1997-1)

=> A = 1997¹⁹⁹⁸ . 1996

Vậy a chia hết cho 4 (vì 1996 chia hết cho 4)

2) B = 1997¹⁹⁹⁸ - 1998¹⁹⁹⁹ 

Mà 1997¹⁹⁹⁸ là số lẻ; 1998¹⁹⁹⁹

=> 1997¹⁹⁹⁸ - 1998¹⁹⁹⁹  là số lẻ

Vậy đề bài sai.

Đặt B = 4²⁰⁰⁴ + 4²⁰⁰³ + 4²⁰⁰² + 4²⁰⁰¹ + ... + 4² + 4 + 1 (có 2005 số; 2005 : 2 dư 1)

B = (4²⁰⁰⁴ + 4²⁰⁰³) + (4²⁰⁰² + 4²⁰⁰¹) + ... + (4² + 4) + 1

B = 4²⁰⁰³.(4 + 1) + 4²⁰⁰¹.(4 + 1) + ... + 4.(4 + 1) + 1

B = 4²⁰⁰³.5 + 4²⁰⁰¹.5 + ... + 4.5 + 1

B = 5.(4²⁰⁰³ + 4²⁰⁰¹ + ... + 4) + 1

=> B = 5 x k + 1 ( k thuộc N*; k chia hết cho 4)

=> A = 75 x (5 x k + 1) + 25

=> A = 75 x 5 x k + 75 + 25

=> A = ...00 + 100

=> A = ..00 chia hết cho 100

A = 75.4^2004 + ... + 75.4 + 75 + 25 
= 25.3.4^2004 + ... + 25.3.4 + 100 
= 100.3.4^2003 + ... + 100.3 + 100 
=> A chia hết cho 100

\(A=75\left(4^{2004}+4^{2003}+....+4+1\right)+25\)

\(\Rightarrow A=75.4^{2004}+75.4^{2003}+....+75.4+75.1+25\)

\(\Rightarrow A=\left(75.4\right).4^{2003}+....+300+100\)

\(\Rightarrow A=300.4^{2003}+.....+300+100\) chia hết cho 100

=> ĐPCM

B=4^2004+4^2003+...+4^2+4+1
Xét 4B = 4^2005+4^2004+...+4^2+4
=> 4B-B = (4^2005+4^2004+...4^3+4^2+4) - (4^2004+4^2003+...+4^2+4+1)
=> 3B = 4^2005 - 1 => B = (4^2005 - 1)/3
=> A = 75 (4^2005 - 1)/3 +25
= 25 (4^2005 -1) +25
= 25 x 4 ^ 2005
= 25 x 4 x 4 ^ 2004 = 100 x4 ^ 2004

\(A=75.(4^{2004}+4^{2003}+...+4^2+4+1)+25\)

Đặt \(B=4^{2004}+4^{2003}+...+4^2+4+1\)

\(4B=4^{2005}+4^{2004}+...+4^3+4^2+4\)

\(4B-B=(4^{2005}+4^{2004}+...+4^3+4^2+4)-\left(4^{2004}+4^{2003}+...+4^2+4+1\right)\)

\(3B=4^{2005}-1\)

\(B=\frac{4^{2005}-1}{3}\)

Thay B vào A ta có

\(A=75.\text{​​}\text{​​}\frac{4^{2005}-1}{3}+25\)

\(A=25.3.(\text{​​}\text{​​}\frac{4^{2005}-1}{3})+25\)

\(A=25.(\text{​​}\text{​​}4^{2005}-1)+25\)

\(A=25.(\text{​​}\text{​​}4^{2005}-1+1)\)

\(A=25.\text{​​}\text{​​}4^{2005}\)

Hok tốt !!!!!!!!!

\(A=75\left(4^{2004}+4^{2003}+4^{2002}+...+4^2+4+1\right)+25\)

\(=75\cdot4^{2004}+75\cdot4^{2003}+75\cdot4^{2002}+...+7\cdot4^2+75\cdot4+\left(75+25\right)\)

\(=3\cdot\left(25\cdot4\right)\cdot4^{2003}+3\cdot\left(25\cdot4\right)\cdot4^{2002}+3\cdot\left(25\cdot4\right)\cdot4^{2001}+...+3\cdot\left(25\cdot4\right)\cdot4+3\cdot\left(25\cdot4\right)+25\cdot4\)

\(=3\cdot100\cdot4^{2003}+3\cdot100\cdot4^{2002}+3\cdot100\cdot4^{2001}+...+3\cdot100\cdot4+3\cdot100+100\)

Mà:

\(3\cdot100\cdot4^{2003}⋮100\)

\(3\cdot100\cdot4^{2002}⋮100\)

\(3\cdot100\cdot4^{2001}⋮100\)

\(...\)

\(3\cdot100⋮100\)

\(100⋮100\)

\(\Rightarrow3\cdot100\cdot4^{2003}+3\cdot100\cdot4^{2002}+3\cdot100\cdot4^{2001}+...+3\cdot100\cdot4+3\cdot100+100⋮100\)

\(\Rightarrow A⋮100\left(đpcm\right)\)