2. Let the integer $k(0 \leqslant k \leqslant 2188)$ be such that 2188 divides $\mathrm{C}_{2188}^{k}$. Find the number of $k$.
2. Note that, $2188=4 \times 547$, and $547$ is a prime number.

If $547 \nmid k$, then
$$
\begin{array}{l}
\mathrm{C}_{2188}^{k}=\frac{2188!}{k!\cdot(2188-k)!} \\
=\frac{2188}{k} \mathrm{C}_{2187}^{k-1}=\frac{4 \times 547}{k} \mathrm{C}_{2187}^{k-1}
\end{array}
$$

is divisible by 547.
If $547 \mid k$, then
$$
k=0,547,1094,1641,2188,
$$

in which case there is no number divisible by 547.
Now, let's verify the cases when $\mathrm{C}_{2188}^{k}$ is odd or divisible by 2 but not by 4.

Let $s(n)$ be the number of 1s in the binary representation of $n$. It is known that the highest power of 2 in the binary decomposition of $n!$ is $n-s(n)$.
Given $2188=(100010001100)_{2}$.
If $\mathrm{C}_{2188}^{k}$ is odd, then
$2188-s(2188)$
$$
\begin{array}{l}
=(k-s(k))+(2188-k-s(2188-k)) \\
\Rightarrow s(2188)=s(k)+s(2188-k) .
\end{array}
$$

This indicates that, in the binary representation, the sum of $k$ and $2188-k$ does not carry over.

Thus, in the binary representation, the positions of 0 in 2188 correspond to 0 in the corresponding positions of $k$; the positions of 1 in 2188 correspond to 0 or 1 in the corresponding positions of $k$. Therefore, there are $2^{4}=16$ ways to choose.
Hence, there are $2189-16=2173$ even numbers in $\mathrm{C}_{2188}^{k}$.
In the binary representation of the 2173 even numbers: the positions of 10 in 2188 (3 groups) correspond to 01 in the corresponding positions of $k$ and $2188-k$; for the other three positions of 1 in 2188, $k$ can be 0 or 1. There are $2^{3}$ ways to choose. Thus,
there are 24 ways in total.
Therefore, there are $2173-24=2149$ numbers that can be divided by 4 or higher powers of 2.
Finally, upon inspection, $\mathrm{C}_{2188}^{0}=\mathrm{C}_{2188}^{2188}=1$ are odd, and $\mathrm{C}_{2188}^{k}(k=547,1094,1641)$ can be divided by $2^{k}\left(k \in \mathbf{Z}_{+}\right.$, $k \geqslant 2)$, so we remove these three numbers.
In summary, the number of $k$ is $2149-3=2146$.