Cho A = 2 +2 mũ 2 +2 mũ 3+ .....+2 mũ 60 Chứng minh A chia hết cho 3;7;15

Giải thích các bước giải:

Ta có:

\(\begin{array}{l}
A = 2 + {2^2} + {2^3} + .... + {2^{60}}\\
 = \left( {2 + {2^2}} \right) + \left( {{2^3} + {2^4}} \right) + \left( {{2^5} + {2^6}} \right) + ..... + \left( {{2^{59}} + {2^{60}}} \right)\\
 = 2.\left( {1 + 2} \right) + {2^3}\left( {1 + 2} \right) + {2^5}\left( {1 + 2} \right) + .... + {2^{59}}\left( {1 + 2} \right)\\
 = \left( {1 + 2} \right)\left( {2 + {2^3} + {2^5} + .... + {2^{59}}} \right)\\
 = 3.\left( {2 + {2^3} + {2^5} + .... + {2^{59}}} \right) \vdots 3\\
A = 2 + {2^2} + {2^3} + .... + {2^{60}}\\
 = \left( {2 + {2^2} + {2^3}} \right) + \left( {{2^4} + {2^5} + {2^6}} \right) + ..... + \left( {{2^{58}} + {2^{59}} + {2^{60}}} \right)\\
 = 2.\left( {1 + 2 + {2^2}} \right) + {2^4}\left( {1 + 2 + {2^2}} \right) + ..... + {2^{58}}\left( {1 + 2 + {2^4}} \right)\\
 = \left( {1 + 2 + {2^2}} \right)\left( {2 + {2^4} + {2^7} + .... + {2^{58}}} \right)\\
 = 7.\left( {2 + {2^4} + {2^7} + .... + {2^{58}}} \right) \vdots 7\\
A = 2 + {2^2} + {2^3} + .... + {2^{60}}\\
 = \left( {2 + {2^2} + {2^3} + {2^4}} \right) + \left( {{2^5} + {2^6} + {2^7} + {2^8}} \right) + .... + \left( {{2^{57}} + {2^{58}} + {2^{59}} + {2^{60}}} \right)\\
 = 2\left( {1 + 2 + {2^2} + {2^3}} \right) + {2^5}\left( {1 + 2 + {2^2} + {2^3}} \right) + ..... + {2^{57}}\left( {1 + 2 + {2^2} + {2^3}} \right)\\
 = \left( {1 + 2 + {2^2} + {2^3}} \right)\left( {2 + {2^5} + .... + {2^{57}}} \right)\\
 = 15.\left( {2 + {2^5} + .... + {2^{57}}} \right) \vdots 15
\end{array}\)