Cmr A=2+2^2+2^3+…+2^20 chia hết cho 5

Đáp án:

Giải thích các bước giải:

$$\eqalign{
& A = 2 + {2^2} + {2^3} + … + {2^{20}} \cr
& \Rightarrow 2A = {2^2} + {2^3} + … + {2^{20}} + {2^{21}} \cr
& \Rightarrow 2A – A = \left( {{2^2} + {2^3} + … + {2^{20}} + {2^{21}}} \right) – \left( {2 + {2^2} + {2^3} + … + {2^{20}}} \right) \cr
& \Rightarrow A = {2^{21}} – 2 = 2\left( {{2^{20}} – 1} \right) \cr
& {2^{20}} = \overline {…76} \Rightarrow {2^{20}} – 1 = \overline {…76} – 1 = \overline {…75} \,\, \vdots \,\,5 \cr
& \Rightarrow 2\left( {{2^{20}} – 1} \right)\,\, \vdots \,\,5 \cr
& \Rightarrow A\,\, \vdots \,\,5 \cr} $$