Đáp án:

$\begin{array}{l}
7)\\
A = 2 + {2^2} + {2^3} + ... + {2^{60}}\\
 = \left( {2 + {2^2} + {2^3} + {2^4}} \right) + ... + \left( {{2^{57}} + {2^{58}} + {2^{59}} + {2^{60}}} \right)\\
 = 2.\left( {1 + 2 + {2^2} + {2^3}} \right) + ... + {2^{57}}\left( {1 + 2 + {2^2} + {2^3}} \right)\\
 = 2.15 + ... + {2^{57}}.15\\
 = \left( {2 + ... + {2^{57}}} \right).15 \vdots 15\\
B = 1 + 3 + {3^2} + ... + {3^{11}}\\
 = \left( {1 + 3} \right) + \left( {{3^2} + {2^3}} \right) + ... + \left( {{3^{10}} + {3^{11}}} \right)\\
 = 4 + {3^2}.4 + ... + {3^{10}}.4\\
 = \left( {1 + {3^2} + ... + {3^{10}}} \right).4 \vdots 4 \Rightarrow B \vdots 4\\
B = \left( {1 + 3 + {3^2}} \right) + \left( {{3^3} + {3^4} + {3^5}} \right) + ... + \left( {{3^9} + {3^{10}} + {3^{11}}} \right)\\
 = 13 + {3^3}.13 + ... + {3^9}.13\\
 = \left( {1 + {3^3} + ... + {3^9}} \right).13 \vdots 13 \Rightarrow B \vdots 13\\
 \Rightarrow B \vdots \left( {4.13} \right) \Rightarrow B \vdots 52\\
c)\\
C = 5 + {5^2} + {5^3} + {5^4} + ... + {5^{12}}\\
 = \left( {5 + {5^2}} \right) + \left( {{5^3} + {5^4}} \right) + ... + \left( {{5^{11}} + {5^{12}}} \right)\\
 = 30 + {5^2}.30 + ... + {5^{10}}.30\\
 = \left( {1 + {5^2} + ... + {5^{10}}} \right).30 \vdots 30\\
C = \left( {5 + {5^2} + {5^3}} \right) + ... + \left( {{5^{10}} + {5^{11}} + {5^{12}}} \right)\\
 = 5\left( {1 + 5 + 25} \right) + ... + {5^{10}}\left( {1 + 5 + 25} \right)\\
 = 5.31 + ... + {5^{10}}.31\\
 = \left( {5 + ... + {5^{10}}} \right).31 \vdots 31\\
8)\\
e){10^6} - {5^7} = {2^6}{.5^6} - {5^6}.5 = {5^6}\left( {{2^6} - 5} \right) = {5^6}.59 \vdots 59\\
g){3^{n + 2}} - {2^{n + 2}} + {3^n} - {2^n}\\
 = {3^n}\left( {{3^2} + 1} \right) - {2^n}\left( {{2^2} + 1} \right)\\
 = {3^n}.10 - {2^n}.5\\
Do:{2^n} \vdots 2 \Rightarrow {2^n}.5 \vdots 10\\
 \Rightarrow {3^n}.10 - {2^n}.5 \vdots 10\\
hay\,{3^{n + 2}} - {2^{n + 2}} + {3^n} - {2^n}\, \vdots \,10
\end{array}$