Ta có: \(C=2^1+2^2+2^3+.........+2^{30}\)
\(=\left(2^1+2^2\right)+\left(2^3+2^4\right)+........+\left(2^{29}+2^{30}\right)\)
\(=2^1\left(1+2\right)+2^3\left(1+2\right)+.........+2^{29}\left(1+2\right)\)
\(=2^1.3+2^3.3+........+2^{99}.3=3.\left(2^1+2^3+.......+2^{99}\right)\)
\(\Rightarrow C⋮3\)
mà \(C=\left(2^1+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+.......+\left(2^{28}+2^{29}+2^{30}\right)\)
\(=2^1\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+.......+2^{28}\left(1+2+2^2\right)\)
\(=2^1.\left(1+2+4\right)+2^4\left(1+2+4\right)+......+2^{28}\left(1+2+4\right)\)
\(=2^1.7+2^4.7+..........+2^{28}.7=7.\left(2^1+2^4+........+2^{28}\right)\)
\(\Rightarrow C⋮7\)
mà \(\left(3;7\right)=1\)\(\Rightarrow C⋮21\)( đpcm )
