\(1,\\ M=\left(4+4^2\right)+\left(4^3+4^4\right)+...+\left(4^{99}+4^{100}\right)\\ M=4\left(4+1\right)+4^3\left(1+4\right)+...+4^{99}\left(4+1\right)\\ M=4\cdot5+4^3\cdot5+...+4^{99}\cdot5\\ M=5\left(4+4^3+...+4^{99}\right)⋮5\\ 2,\\ M=1+\left(3+3^2\right)+\left(3^3+3^4\right)+...+\left(3^{2019}+3^{2020}\right)\\ M-1=3\left(1+3\right)+3^3\left(1+3\right)+...+3^{2019}\left(1+3\right)\\ M-1=3\cdot4+3^3\cdot4+...+3^{2019}\cdot4\\ M-1=4\left(3+3^3+...+3^{2019}\right)⋮4\)
\(3,\\ B=2+2^2+2^3+...+2^{30}\\ B=\left(2+2^2\right)+\left(2^3+2^4\right)+...+\left(2^{29}+2^{30}\right)\\ B=2\left(2+1\right)+2^3\left(2+1\right)+...+2^{29}\left(2+1\right)\\ B=3\left(2+2^3+...+2^{29}\right)⋮3\left(1\right)\\ B=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)...+\left(2^{28}+2^{29}+2^{30}\right)\\ B=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{28}\left(1+2+2^2\right)\\ B+7\left(2+2^4+...+2^{28}\right)⋮7\left(2\right)\\ \left(1\right)\left(2\right)\Rightarrow B⋮21\)
\(4,\\ a,\overline{x138y}⋮2;\overline{x138y}⋮5\\ \Rightarrow\overline{x138y}⋮10\Rightarrow y=0\\ \overline{x1380}⋮9\Rightarrow x+1+3+8+0⋮9\\ \Rightarrow x+12⋮9\Rightarrow x=6\\ \Rightarrow\overline{x138y}=61380\\ b,\overline{x138y}:2.dư.3\\ \overline{x138y}:5.dư.3\\ \Rightarrow\overline{x138y}:10.dư.3\\ \Rightarrow y=3\\ \overline{x138y}:9.dư.3\\ \Rightarrow\overline{x1383}:9.dư.3\\ \Rightarrow x+1+3+8+3:9.dư.3\\ \Rightarrow x+15:9.dư.3\\ \Rightarrow x=6\\ \Rightarrow\overline{x138y}=61383\)
