a) \(S=5+5^2+5^3+...+5^{96}\)
\(S=\left(5+5^2+5^3+5^4+5^5+5^6\right)+...+\left(5^{91}+5^{92}+5^{93}+5^{94}+5^{95}+5^{96}\right)\)
\(S=5.\left(1+5+5^2+5^3+5^4+5^5\right)+...+5^{91}.\left(1+5^2+5^3+5^4+5^5\right)\)
\(S=5.3906+...+5^{91}.3906\)
\(S=3906.\left(5+...+5^{96}\right)\)
\(S=3.126.\left(5+...+5^{91}\right)\) chia hết cho \(6.\)
b) Do \(S\) là tổng các lũy thừa có cơ số là \(5\).
Cho nên mỗi lũy thừa đều tận cùng là \(5\).
Mà \(S\) có tất cả \(96\) số
\(\Rightarrow\) Chữ số tận cùng của \(S\) là \(0\).
\(S=5+5^2+5^3+..+5^{96}\)
\(S=\left(5+5^2+5^3+5^4+5^5+5^6\right)+\left(5^7+5^8+5^9+5^{10}+5^{11}+5^{12}\right)+...+\left(5^{91}+5^{92}+5^{93}+5^{94}+5^{95}+5^{96}\right)\)\(S=1\left(5+5^2+5^3+5^4+5^6\right)5^6\left(5+5^2+5^3+5^4+5^5+5^6\right)+...+5^{90}+\left(5+5^2+5^3+5^4+5^5+5^6\right)\)\(S=\left(5+5^2+5^3+5^4+5^5+5^6\right)\left(1+5^6+...+5^{90}\right)\)\(S=19530\left(1+5^6+...+5^{90}\right)\)
\(S=155.126.\left(1+5^6+...+5^{90}\right)\)
\(S⋮126\rightarrowđpcm\)
\(S=5+5^2+5^3+...+5^{96}\)
\(S=\overline{...5}+\overline{...5}+\overline{...5}+\overline{...5}+...+\overline{...5}+\overline{...5}\)\(S=\left(\overline{...5}+\overline{...5}\right)+\left(\overline{...5}+\overline{...5}\right)+...+\left(\overline{...5}+\overline{...5}\right)\)\(S=\overline{...0}+\overline{...0}+\overline{...0}\)
\(S=\overline{...0}\)
