a) Ta có:
\(M=1+3+3^2+3^3+...+3^{999}\)
\(M=\left(1+3\right)+\left(3^2+3^3\right)+...+\left(3^{998}+3^{999}\right)\)
\(M=4+3^2.\left(1+3\right)+...+3^{998}.\left(1+3\right)\)
\(M=4+3^2.4+...+3^{998}.4\)
\(M=\left(1+3^2+...+3^{998}\right).4⋮4\)
\(\Rightarrow M⋮4\)
c) \(M=1+3+3^2+...+3^{999}\)
\(\Rightarrow3M=3+3^2+3^3+...+3^{1000}\)
\(\Rightarrow3M-M=\left(3+3^2+3^3+...+3^{1000}\right)-\left(1+3+3^2+3^3+...+3^{999}\right)\)
\(\Rightarrow2M=3^{1000}-1\)
\(\Rightarrow M=\frac{3^{1000}-1}{2}\)
b) \(M=1+3+3^2+3^3+...+3^{999}\)
\(M=\left(1+3+3^2\right)+\left(3^3+3^4+3^5\right)+...+\left(3^{997}+3^{998}+3^{999}\right)\)
\(M=15+3^3.\left(1+3+3^2\right)+...+3^{997}.\left(1+3+3^2\right)\)
\(M=15+3^3.15+...+3^{997}.15\)
\(M=\left(1+3^3+...+3^{997}\right).15\)
Vì 15 chia 13 có số dư là 2 nên M chia 13 có số dư là 2
Vậy M chia 13 dư 2