5. From $26=1^{2}+5^{2}=1^{2}+3^{2}+4^{2}$, it can be concluded that 26 can be represented as the sum of squares of at most 3 distinct non-zero natural numbers. Please determine the maximum number of distinct non-zero natural numbers whose squares can sum up to 360?
5.【Solution】List the squares of numbers from 1 to 18:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline & $1 \mid 3$ & & & 6 & 7 & 8 & 9 & 10 & 111 & $12 \|$ & $\mid 13$ & $\mid 14$ & 15 & 16 & 17 & \\
\hline & & & & & & & & 0 & 121 & 144 & 169 & & 225 & 256 & 289 & \\
\hline
\end{tabular}

To get the sum of as many squares as possible, start with the smallest numbers, beginning from $1^{2}$:
$1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2}=385$, which is already greater than 360, and it is exactly 25 more than 360, where $25=5^{2}$.

Therefore, 360 can be expressed as the sum of the squares of 9 distinct non-zero natural numbers, which is:
$$
360=1^{2}+2^{2}+3^{2}+4^{2}+6^{2}+7^{2}+8^{2}+9^{2}+10^{2}
$$