\(A=5+5^3+5^5+...+5^{239}\)

\(=\left(5+5^3\right)+\left(5^5+5^7\right)+...+\left(5^{237}+5^{239}\right)\)

\(=5\left(1+5^2\right)+5^5\left(1+5^2\right)+...+5^{237}\left(1+5^2\right)\)

\(=\left(1+5^2\right)\left(1+5^5+..+5^{237}\right)\)

\(=26\left(1+5^5+...+5^{237}\right)\)

Vì  26 chia hết cho 13 nên    \(26\left(1+5^5+...+5^{237}\right)\)chia hết cho 13

Vậy A chia hết cho 13