Ta có             \(2^{13}.5^8.10\\ \\ =2^8.2^5.5^8.10\\ \\ =\left(2^8.5^8\right).2^5.10\\ \\ =10^8.10.32\\ \\ =10^9.32\\ \\ =32000000000\)⇒  \(2^{13}.5^8.10\) có 11 chữ số

\(a=2^{13}.5^8.10\)

\(a=2^8.2^5.5^8.10\)

\(a=\left(2^8.5^8\right).2^5.10\)

\(a=\left(2.5\right)^8.32.10\)

\(a=10^8.32.10\)

\(a=1000...0\left(8cs0\right).10.32\)

\(a=1000...0\left(9cs0\right).32\)

\(a=32000...0\left(9cs0\right)\)

\(\Rightarrow a\) có 11 chữ số

1. \(x-8+32=68\)

\(x-8=68-32\)

\(x-8=36\)

\(x=36+8\)

\(x=44\)

_____

2. \(x+8+32=68\)

\(x+8=68-32\)

\(x+8=36\)

\(x=36-8\)

\(x=28\)

_____

3. \(98-x+34=43\)

\(98-x=43-34\)

\(98-x=9\)

\(x=98-9\)

\(x=89\)

_____

4. \(98+x-34=43\)

\(98+x=43+34\)

\(98+x=77\)

\(x=77-98\)

\(x=-21\)

_____

5. \(x:5:4=800\)

\(x:5=800.4\)

\(x:5=3200\)

\(x=3200.5\)

\(x=16000\)

_____

6. \(18+x=384:8\)

\(18+x=48\)

\(x=48-18\)

\(x=30\)

_____

7. \(\left(84,6-2\times x\right):3,02=5,1\)

\(84,6-2.x=5,1.3,02\)

\(84,6-2.x=15,402\)

\(2.x=84,6-15,402\)

\(2.x=69,198\)

\(x=69,198:2\)

\(x=34,599\)

_____

8. \(x\times5=120:6\)

\(x.5=20\)

\(x=20:5\)

\(x=4\)

_____

9. \(\left(15\times24-x\right):0,25=100:0,25\)

\(\left(15.24-x\right):0,25=400\)

\(15.24-x=400.0,25\)

\(15.24-x=100\)

\(360-x=100\)

\(x=360-100\)

\(x=260\)

1,x-8+32=68

=>x-8=68-32=36

=>x=36+8=42

2,x+8+32=68

=>x+8=68-32=36

=>x=36-8=28

3,98-x+34=43

=>98-x=43-34=9

=>x=98-9=89

4,98+x-34=43

=>98+x=43+34=77

=>x=77-98=-21

5,x:5:4=800

=>x:20=800

=>x=800x20=16000

6,18+x=384:8=48

=>x=48-18=30

7,(84,6-2x):3,02=5,1

=>84,6-2x=5,1x3,02=15,402

=>2x=84,6-15,402=69,198

=>x=69,198:2=34,599

8,5x=120:6=20

=>x=20:5=4

9,(15x24-x):0,25=100:0,25=400

=>360-x=400x0,25=100

=>x=360-100=260.

Ta có a = 2¹³ . 5⁸ . 10

             = 2⁸ . 2⁵ . 5⁸ . 10

             = 10⁸ . 10 . 32

             = 10⁹ . 32

             = 1 000 000 000 . 32

             = 32 000 000 000

Vậy a có 11 chữ số

11

(3x-15)x3⁷=3⁸

=>3x-15=3⁸:3⁷=3

=>3x=3+15=18

=>x=18:3=6.

\(\left(3x-15\right)\cdot3^7=3^8\\ \\ \Rightarrow3x-15=3^8:3^7\\ \\ \Rightarrow3x-15=3^{8-7}=3^1\\ \\ \Rightarrow3x-15=3\Rightarrow3x=3+15=18\\ \\ \Rightarrow x=18:3=6\)

\(2VT=2^{x+1}+2^{x+2}+2^{x+3}+...+...+2^{x+2016}\)

\(VT=2VT-VT=2^{x+2016}-2^x=2^{2016}.2^x+2^x=2^x\left(2^{2016}+1\right)\)

\(VP=2^{2019}-2^3=2^3\left(2^{2016}-1\right)\)

\(\Rightarrow2^2\left(2^{2016}-1\right)=2^3\left(2^{2016}-1\right)\)

\(\Rightarrow2^x=2^3\Rightarrow x=3\)

\(2^x+2^{x+1}+2^{x+2}+2^{x+2015}=2^{2019}-8\left(1\right)\)

Đặt \(S=2^x+2^{x+1}+2^{x+2}+2^{x+2015}\)

\(\Rightarrow S+\left(1+2^2+...2^{x-1}\right)=\left(1+2^2+...2^{x-1}\right)+2^x+2^{x+1}+2^{x+2}+2^{x+2015}\)

\(\Rightarrow S+\dfrac{2^{x-1+1}-1}{2-1}=1+2^2+...2^{x-1}+2^x+2^{x+1}+2^{x+2}+2^{x+2015}\)

\(\Rightarrow S+2^x-1=\dfrac{2^{x+2015+1}-1}{2-1}\)

\(\Rightarrow S+2^x-1=2^{x+2016}-1\)

\(\Rightarrow S=2^{x+2016}-2^x\)

\(\left(1\right)\Rightarrow2^{x+2016}-2^x=2^{2019}-8=2^{2019}-2^3\)

\(\Rightarrow2^x\left(2^{2016}-1\right)=2^3\left(2^{2016}-1\right)\)

\(\Rightarrow2^x=2^3\Rightarrow x=3\)

Sửa câu a

a)Ta có:

\(A=3+3^2+3^3+...+3^{99}\)

 \(A=\left(3+3^2+3^3\right)+...+\left(3^{97}+3^{98}+3^{99}\right)\) 

\(A=\left(3+3^2+3^3\right)+...+3^{96}.\left(3+3^2+3^3\right)\)

\(A=39+...+3^{96}.39\)

\(A=39.\left(1+...+3^{96}\right)\)

Vì 39 \(⋮\) 13 nên 39 . ( 1 + ... + 3⁹⁶ ) \(⋮\) 13

Vậy A \(⋮\) 13

_________

b)Ta có:

 \(B=5+5^2+5^3+...+5^{50}\)

\(B=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{49}+5^{50}\right)\)

\(B=\left(5+5^2\right)+5^2.\left(5+5^2\right)+...+5^{48}.\left(5+5^2\right)\)

\(B=30+5^2.30+...+5^{48}.30\)

\(B=30.\left(1+5^2+...+5^{48}\right)\)

Vì 30 \(⋮\) 6 nên 30. ( 1 + 5² + ... + 5⁴⁸ ) \(⋮\) 6

Vậy B \(⋮\) 6

a,A=3+3²+3³+..+3⁹⁹=(3+3²+3³)+...+(3⁹⁷+3⁹⁸+3⁹⁹)

     =3(1+3+3²)+...+3⁹⁷(1+3+3²)=3x13+...+3⁹⁷x13=13(3+...+97)⋮13

b,B=5+5²+...+5⁵⁰=(5+5²)+...+(5⁴⁹+5⁵⁰)=5(1+5)+..+5⁴⁹(1+5)

  =5x6+...+5⁴⁹x6=6(5+..+5⁴⁹)⋮6.

31¹¹<32¹¹=2⁵⁵

17¹⁴>16¹⁴=2⁵⁶

=>Vì 2⁵⁵<2⁵⁶ nên 31¹¹<17¹⁴.

\(...\Rightarrow\left[{}\begin{matrix}288:\left(x-3\right)^2-2=0\\x^2-169=0\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}288:\left(x-3\right)^2=2\\x^2=169\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}\left(x-3\right)^2=288:2\\x^2=13^2\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}\left(x-3\right)^2=144=12^2\\x^2=13^2\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x-3=12\\x-3=-12\\x=13\\x=-13\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=15\\x=-9\\x=13\\x=-13\end{matrix}\right.\) \(\Rightarrow x\in\left\{-9;15;\pm13\right\}\)

\(3n+2⋮n-1\)

\(\Rightarrow3n+2-3\left(n-1\right)⋮n-1\)

\(\Rightarrow3n+2-3n+3⋮n-1\)

\(\Rightarrow5⋮n-1\)

\(\Rightarrow n-1\in\left\{-1;1;-5;5\right\}\)

\(\Rightarrow n\in\left\{0;2;-4;6\right\}\left(n\in Z\right)\)

Số dư lớn nhất trong 1 phép chia bằng số chia -1

=> số dư lớn nhất trong phép chia trên = 5-1=4

Theo đề bài số dư = {2;4}

Với số dư = 2 thì thương là 2:2=1

=>x=5x1+2=7

Với số dư = 4 thì thương là 4:2=2

=> x=5x2+4=14

Theo đề :

\(x=5.r+2.r\)

mà \(2.r< 5\Rightarrow r\in\left\{1;2\right\}\Rightarrow x\in\left\{7;14\right\}\)