\(A=4+4^2+4^3+...+4^{23}+4^{24}\)

\(=\left(4+4^2\right)+\left(4^3+4^4\right)+...+\left(4^{23}+4^{24}\right)\)

\(=20+4^3.\left(4+4^2\right)+....+4^{23}.\left(4+4^2\right)\)

\(=1.20+4^3.20+....+4^{23}.20\)

\(=\left(1+4^3+...+4^{23}\right).20\)

\(\Rightarrow A⋮20\)

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\(A=4+4^2+4^3+....+4^{23}+4^{24}\)

\(=\left(4+4^2+4^3\right)+\left(4^4+4^5+4^6\right)+....+\left(4^{22}+4^{23}+4^{24}\right)\)

\(=84+4^4.\left(4+4^2+4^3\right)+.....+4^{22}.\left(4+4^2+4^3\right)\)

\(=1.84+4^4.84+....+4^{22}.84\)

\(=\left(1+4^4+...+4^{22}\right).84\)

\(\Rightarrow A⋮84⋮21\)

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\(A=4+4^2+4^3+......+4^{23}+4^{24}\)\(=\left(4+4^2+4^3+4^4+4^5+4^6\right)+\left(4^7+4^8+4^9+4^{10}+4^{11}+4^{12}\right)+...+\left(4^{19}+4^{20}+4^{21}+4^{22}+4^{23}+4^{24}\right)\)

\(=5460+4^7.\left(4+4^2+4^3+4^4+4^5+4^6\right)+....+4^{19}.\left(4+4^2+4^3+4^4+4^5+4^6\right)\)

\(=1.5460+4^7.5460+...4^{19}.5460\)

\(=\left(1+4^7+...+4^{19}\right).5460\)

\(\Rightarrow A⋮5460⋮420\)

:0

chi mà giỏi zữ zayyyyyy;0

\(A=\left(3+3^2+3^3\right)+...+\left(3^{58}+3^{59}+3^{60}\right)\\ A=3\left(1+3+3^2\right)+...+3^{58}\left(1+3+3^2\right)\\ A=\left(1+3+3^2\right)\left(3+...+3^{58}\right)\\ A=13\left(3+...+3^{58}\right)⋮13\)

\(M=\left(2+2^2+2^3+2^4\right)+...+\left(2^{17}+2^{18}+2^{19}+2^{20}\right)\\ M=\left(2+2^2+2^3+2^4\right)+...+2^{16}\left(2+2^2+2^3+2^4\right)\\ M=\left(2+2^2+2^3+2^4\right)\left(1+...+2^{16}\right)\\ M=30\left(1+...+2^{16}\right)⋮5\)

+ Do n không chia hết cho 3 => 4n không chia hết cho 3; 3 chia hết cho 3 => 4n + 3 không chia hết cho 3 => (4n + 3)² không chia hết cho 3

=> (4n + 3)² chia 3 dư 1 (1)

+ Do 4n + 3 lẻ => (4n + 3)² lẻ => (4n + 3)² chia 8 dư 1 (2)

Từ (1) và (2); do (3;8)=1 => (4n + 3)² chia 24 dư 1

Mà 25 chia 24 dư 1

=> (4n + 3)² - 25 chia hết cho 24 ( đpcm)

\(S=3^1+3^2+3^3+.....+3^{100}\) \(=\left(3^1+3^2+3^3+3^4\right)+\left(3^5+3^6+3^7+3^8\right)+...+\left(3^{97}+3^{98}+3^{99}+3^{100}\right)\)

\(=120+3^5.\left(3^1+3^2+3^3+3^4\right)+....+3^{97}.\left(3^1+3^2+3^3+3^4\right)\)

\(=1.120+3^5.120+...+3^{97}.120\)

\(=\left(1+3^5+...+3^{97}\right).120\)

\(\Rightarrow S⋮120\)

Vậy ........

a) Thay m = -1 và n = 2 ta có:

3m - 2n = 3(-1) -2.2 = -3 - 4 = -7

b) Thay m = -1 và n = 2 ta được 

7m + 2n - 6 = 7.(-1) + 2.2 - 6 = -7 + 4 - 6 = -9.

kp[