5. A.

The original equation can be transformed to
$$
(2 x-13)(2 y-13)=169 \text {. }
$$

Since $x \neq y$, then
$$
\begin{array}{l}
\left\{\begin{array} { l } 
{ 2 x - 1 3 = 1 , } \\
{ 2 y - 1 3 = 1 6 9 }
\end{array} \Rightarrow \left\{\begin{array}{l}
x=7, \\
y=91 .
\end{array}\right.\right. \\
\text { Therefore, } x^{y}+y^{x}=7^{91}+91^{7} \\
\equiv\left(7^{4}\right)^{22} \times 7^{3}+(-9)^{7} \equiv 7^{3}-81^{3} \times 9 \\
\equiv 43-41 \times 9 \equiv-26 \equiv 74(\bmod 100) .
\end{array}
$$
5. Let $x$ and $y$ be distinct positive integers, satisfying $\frac{1}{x}+\frac{1}{y} = \frac{2}{13}$. Then the last two digits of $x^{y}+y^{x}$ are ( ).
(A) 74
(B) 84
(C) 75
(D) 85