\(M=2+2^2+2^3+...+2^{20}\\ =\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{17}+2^{18}+2^{19}+2^{20}\right)\\ =1\cdot\left(2+2^2+2^3+2^4\right)+2^4\cdot\left(2+2^2+2^3+2^4\right)+...+2^{16}\cdot\left(2+2^2+2^3+2^4\right)\\ =1\cdot30+2^4\cdot30+...+2^{16}\cdot30\\ =30\cdot\left(1+2^4+...+2^{16}\right)\\ =3\cdot10\cdot\left(1+2^4+...+2^{16}\right)⋮10\left(đpcm\right)\)

\(M=2+2^2+2^3+...+2^{20}\\ =\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{17}+2^{18}+2^{19}+2^{20}\right)\\ =1\cdot\left(2+2^2+2^3+2^4\right)+2^4\cdot\left(2+2^2+2^3+2^4\right)+...+2^{16}\cdot\left(2+2^2+2^3+2^4\right)\\ =1\cdot30+2^4\cdot30+...+2^{16}\cdot30\\ =30\cdot\left(1+2^4+...+2^{16}\right)⋮30\left(đpcm\right)\)

\(a,1^3+2^3=1+8=9\\ b,1^3+2^3+3^3=1+8+27=9+27=36\\ c,1^3+2^3+3^3+4^3=1+8+27+64=9+91=100\\ d,\left(2^{13}+2^5\right):\left(2^{10}+2^2\right)=2^3\left(2^{10}+2^2\right):\left(2^{10}+2^2\right)=2^3=8\)

\(a,2^x=16\\ \Rightarrow2^x=2^4\\ \Rightarrow x=4\\ b,4^x=64\\ \Rightarrow4^x=4^3\\ \Rightarrow x=3\\ c,15^x=225\\ \Rightarrow15^x=15^2\\ \Rightarrow x=2\\ d,3^x\cdot3=243\\ \Rightarrow3^x=81\\ \Rightarrow3^x=3^4\\ \Rightarrow x=4\\ e,2^x\cdot7=56\\ \Rightarrow2^x=8\\ \Rightarrow2^x=2^3\\ \Rightarrow x=3\\ g,x^6:x^3=125\\ \Rightarrow x^3=125\\ \Rightarrow x^3=5^3\\ \Rightarrow x=5\)

\(\left(x+2y+3z\right)\left(2y+3z-x\right)\\= \left[\left(2y+3z\right)+x\right]\left[\left(2y+3z\right)-x\right]\\ =\left(2y+3z\right)^2-x^2\\=4y^2+12yz+9z^2-x^2\)

\(\left(y+2z-3\right)\left(y-2z-3\right)\\ =\left[\left(y-3\right)+2z\right]\left[\left(y-3\right)-2z\right]\\ =\left(y-3\right)^2-\left(2z\right)^2\\ =y^2-6y+9-4z^2\)

\(\rightarrow\) Số lớn khác nhau \(⋮5\) dư \(2\) là: \(102\)

Để \(2n-1⋮n-1\), ta có:

\(2n-1⋮n-1\\ \Rightarrow2n-2+1⋮n-1\\ \Rightarrow2\left(n-1\right)+1⋮n-1\)

Vì: \(2\left(n-1\right)⋮n-1\rightarrow1⋮n-1\rightarrow n-1\inƯ\left(1\right)=\left\{1\right\}\)

\(\Rightarrow n=2\)

Vậy: \(n=2\) thì \(2n-1⋮n-1\)

\(x:11+8712=8771\\ x:11=8771-8712\\ x:11=59\\ x=59\times11\\ x=649\)

\(a,\) Ta có:

\(3^{484}=3^{4\cdot121}=\left(3^4\right)^{121}=81^{121}\\ 4^{363}=4^{3\cdot121}=\left(4^3\right)^{121}=64^{121}\)

Vì: \(81^{121}>64^{121}\rightarrow3^{484}>4^{363}\)

\(b,\) Ta có:

\(5^{300}=5^{2\cdot150}=\left(5^2\right)^{150}=25^{150}\\ 3^{453}=3^{3\cdot151}=\left(3^3\right)^{151}=27^{151}=27\cdot27^{150}\)

Vì: \(25^{150}< 27\cdot27^{150}\rightarrow5^{300}< 3^{453}\)