A=\(\text{2}^{1}+\text{2}^{2}+\text{2}^{3}+\text{2}^{4}+...+\text{2}^{2016}\)
=\((\text{2}^{1}+\text{2}^{2})+(\text{2}^{3}+\text{2}^{4})+...+(\text{2}^{2015}+\text{2}^{2016})\)
=\(2.(1+2)+\text{2}^{3}(1+2)+...+\text{2}^{2015}(1+2)\)
=\((2+\text{2}^{3}+\text{2}^{5}+...+\text{2}^{2015}).(1+2)\)
=\((2+\text{2}^{3}+\text{2}^{5}+...+\text{2}^{2015}).3\)⋮\(3\)
Vậy A⋮3
A=\(\text{2}^{1}+\text{2}^{2}+\text{2}^{3}+\text{2}^{4}+...+\text{2}^{2016}\)
=\((\text{2}^{1}+\text{2}^{2}+\text{2}^{3})+(\text{2}^{4}+\text{2}^{5}+\text{2}^{6})+...+(\text{2}^{2014}+\text{2}^{2015}+\text{2}^{2016})\)
=\(2(1+\text{2}^{1}+\text{2}^{2})+\text{2}^{4}(1+\text{2}^{1}+\text{2}^{2})+...+\text{2}^{2014}(1+\text{2}^{1}+\text{2}^{2})\)
=\((2+\text{2}^{4}+...+\text{2}^{2014})(1+\text{2}^{1}+\text{2}^{2})\)
=\((2+\text{2}^{4}+...+\text{2}^{2014})7\)⋮\(7\)
Vậy A⋮7
