Example 3 Can 2010 be written as the sum of squares of $k$ distinct positive integers? If so, try to find the maximum value of $k$; if not, please briefly explain the reason. ${ }^{[2]}$
(2010, Beijing Middle School Mathematics Competition (Grade 8))
Estimate the approximate range of $k$ first, then discuss by classification.
Let $p_{i}$ be a prime number.
If 2010 can be written as the sum of squares of $k$ prime numbers, then by
$$
\begin{array}{l}
2^{2}+3^{2}+5^{2}+7^{2}+11^{2}+13^{2}+17^{2}+19^{2}+23^{2}+29^{2} \\
=2397>2010,
\end{array}
$$

we know $k \leqslant 9$.
(1) When $k=9$, if
$$
2010=p_{1}^{2}+p_{2}^{2}+\cdots+p_{9}^{2} \text {, }
$$

among them, there must be one square of an even number and eight squares of odd prime numbers.

The left side of the equation is congruent to 2 modulo 8, while the right side is congruent to 4 modulo 8, so the equation cannot hold.
(2) When $k=8$, if
$$
2010=p_{1}^{2}+p_{2}^{2} \cdots+p_{8}^{2} \text {, }
$$

these eight addends are all squares of odd prime numbers.

The left side of the equation is congruent to 2 modulo 8, while the right side is congruent to 0 modulo 8, so the equation cannot hold.
(3) When $k=7$, after trial calculation, we find
$$
2^{2}+3^{2}+7^{2}+11^{2}+13^{2}+17^{2}+37^{2}=2010 \text {. }
$$

In summary, the maximum value of $k$ is 7.