13. Find the integer which is closest to $\frac{(1+\sqrt{3})^{4}}{4}$.
13. Answer: 14
$$
\begin{aligned}
(1+\sqrt{3})^{2}=1^{2}+3+2 \sqrt{3} & =4+2 \sqrt{3} \\
(1+\sqrt{3})^{4}=\left((1+\sqrt{3})^{2}\right)^{2}=(4+2 \sqrt{3})^{2} & =4^{2}+(2 \sqrt{3})^{2}+2 \cdot 4 \cdot 2 \sqrt{3} \\
& =16+12+16 \sqrt{3} \\
& =28+16 \sqrt{3}
\end{aligned}
$$

Hence,
$$
\frac{(1+\sqrt{3})^{4}}{4}=\frac{28+16 \sqrt{3}}{4}=7+4 \sqrt{3} .
$$

Note that $1.7<\sqrt{3}<1.8$. Thus,
$$
\begin{aligned}
& 7+4 \times 1.7<7+4 \sqrt{3}<7+4 \times 1.8 \\
\Longrightarrow \quad & 13.8<7+4 \sqrt{3}<14.2 .
\end{aligned}
$$

Therefore, the integer closest to $\frac{(1+\sqrt{3})^{4}}{4}$ is 14 .