a: \(A=2\left(1+2\right)+2^3\left(1+2\right)+...+2^{99}\left(1+2\right)\)

=3(2+2^3+...+2^99) chia hết cho 3

b: Sửa đề: \(B=3+3^2+3^3+...+3^{1990}+3^{1991}+3^{1992}\)

\(=3\left(1+3+3^2\right)+...+3^{1990}\left(1+3+3^2\right)\)

\(=13\left(3+...+3^{1990}\right)⋮13\)

\(S1=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{99}+5^{100}\right)\)

\(=5.\left(1+5\right)+5^3.\left(1+5\right)+...+5^{99}.\left(1+5\right)\)

\(=5.6+5^3.6+...+5^{99}.6\)

\(=6.\left(5+5^3+...+5^{99}\right)⋮6\)

câu b tương tự

\(S3=16^5+21^5\)

vì 16+21=33 chia hết cho 33

=>16⁵+21⁵ chia hết cho 33

P/S: theo công thức:(n+m chia hết cho a=> n^b+m^b chia hết cho a)

S1 = 5+5²+5³+...+5⁹⁹+5¹⁰⁰

=5. (1+5)+5³ . (1+5) + ... + 5⁹⁹.(1+5)

= 5.6 +5³.6+..+ 5⁹⁹.6

=6.(5+5³ + ... +5⁹⁹):6

vậy x = ...

b)2+2²+2³+...+2⁹⁹+2¹⁰⁰

=2.(1+2)+2³.(1+2) + ... + 2⁹⁹.(1+2)

=2.3+2³+..+2⁹⁹):3

= ....

c)16⁵+21⁵

vì 16+21 chia hế 33 nên

theo công thức(n+m chia hết cho a=(n^b+m^b)

S1 = 5+5²+5³+...+5⁹⁹+5¹⁰⁰

=5. (1+5)+5³ . (1+5) + ... + 5⁹⁹.(1+5)

= 5.6 +5³.6+..+ 5⁹⁹.6

=6.(5+5³ + ... +5⁹⁹):6

vậy x = ...

b)2+2²+2³+...+2⁹⁹+2¹⁰⁰

=2.(1+2)+2³.(1+2) + ... + 2⁹⁹.(1+2)

=2.3+2³+..+2⁹⁹):3

= ....

c)16⁵+21⁵

vì 16+21 chia hế 33 nên

theo công thức(n+m chia hết cho a=(n^b+m^b)

(1+2³)+(2+2⁴)+...+(2⁸+2¹¹)

9+2(1+2³)+...+2⁸(1+2³)

9(1+2+...+2⁸) chia hết cho 9

=>( 2^0+2^1+2^2 + ...+2^11) chia hết cho 9

c)(5+5²)+(5³+5⁴)+...+(5⁹⁹+5¹⁰⁰)

5(1+5)+5³(1+5)+...+5⁹⁹(1+5)

6(5+5³+...+5⁹⁹) chia hết cho 3

a) 5+5²+5³+5⁴+...+5¹⁰⁰

= (5+5²)+(5³+5⁴)+...+(5⁹⁹+5¹⁰⁰)

= 30+5².(5+5²)+...+5⁹⁸.(5+5²)

= 30+5².30+...+5⁹⁸.30

= 30.(1+5²+...+5⁹⁸)

Vì 30 chia hết cho 10

=> 30.(1+5²+...+5⁹⁸) chia hết cho 10

=> 5+5²+5³+...+5¹⁰⁰ chia hết cho 10

1.

\(\left(x+2\right)^3=\frac{1}{8}\)

\(\Rightarrow\left(x+2\right)^3=\left(\frac{1}{2}\right)^3\)

\(\Rightarrow x+2=\frac{1}{2}\)

\(\Rightarrow x=\frac{1}{2}-2\)

\(\Rightarrow x=-\frac{3}{2}\)

Vậy \(x=-\frac{3}{2}.\)

2.

b) Ta có:

\(5^5-5^4+5^3\)

\(=5^3.\left(5^2-5+1\right)\)

\(=5^3.\left(25-5+1\right)\)

\(=5^3.21\)

Vì \(21⋮7\) nên \(5^3.21⋮7.\)

\(\Rightarrow5^5-5^4+5^3⋮7\left(đpcm\right).\)

c) Ta có:

\(2^{19}+2^{21}+2^{22}\)

\(=2^{19}.\left(1+2^2+2^3\right)\)

\(=2^{19}.\left(1+4+8\right)\)

\(=2^{19}.13\)

Vì \(13⋮13\) nên \(2^{19}.13⋮13.\)

\(\Rightarrow2^{19}+2^{21}+2^{22}⋮13\left(đpcm\right).\)

Chúc bạn học tốt!

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