How many integers between $1000$ and $2000$ have all three of the numbers $15$, $20$, and $25$ as factors?

$\textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 3 \qquad
\textbf{(D)}\ 4 \qquad
\textbf{(E)}\ 5$
1. To determine how many integers between $1000$ and $2000$ are divisible by $15$, $20$, and $25$, we first need to find the least common multiple (LCM) of these three numbers.
2. We start by finding the prime factorizations of $15$, $20$, and $25$:
   \[
   15 = 3 \times 5
   \]
   \[
   20 = 2^2 \times 5
   \]
   \[
   25 = 5^2
   \]
3. The LCM is found by taking the highest power of each prime that appears in the factorizations:
   \[
   \text{LCM}(15, 20, 25) = 2^2 \times 3 \times 5^2
   \]
4. Calculating the LCM:
   \[
   2^2 = 4
   \]
   \[
   5^2 = 25
   \]
   \[
   4 \times 3 = 12
   \]
   \[
   12 \times 25 = 300
   \]
   Therefore, the LCM of $15$, $20$, and $25$ is $300$.
5. Next, we need to find the integers between $1000$ and $2000$ that are multiples of $300$.
6. The smallest multiple of $300$ greater than $1000$ is:
   \[
   300 \times 4 = 1200
   \]
7. The largest multiple of $300$ less than $2000$ is:
   \[
   300 \times 6 = 1800
   \]
8. The multiples of $300$ between $1000$ and $2000$ are:
   \[
   1200, 1500, 1800
   \]
9. Therefore, there are $3$ integers between $1000$ and $2000$ that are divisible by $15$, $20$, and $25$.

The final answer is $\boxed{3}$.