Problem 460

Manipulating the left side, $2^{11} \times 6^5 = 2^{11} \times (2 \times 3)^5 = 2^{11} \times 2^5 \times 3^5 = 2^{16} \times 3^5$.

Since $4^x \times 3^y = 2^{16} \times 3^5$ and $x$ and $y$ are positive integers, then $y = 5$ (because $4^x$ has no factors of 3).

This also means that $4^x = 2^{16}$.

Since $4^x = (2^2)^x = 2^{2x}$, then
$4^x = 2^{16}$ gives $2^{2x} = 2^{16}$ and so $2x = 16$ or $x=8$.

Therefore, $x + y = 8 + 5 = 13$.