In order for Timothy's quotient to be an integer, the integer Josef picks must be a divisor of 1000.
To find the number of divisors of 1000, we can prime factorize 1000 as $2^3 \cdot 5^3$.
Any divisor of 1000 can be expressed as $2^a \cdot 5^b$, where $0 \leq a \leq 3$ and $0 \leq b \leq 3$.
There are 4 choices for $a$ and 4 choices for $b$, so there are $4 \cdot 4 = \boxed{16}$ integers Josef can pick.
The answer is: 16
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Josef and Timothy play a game in which Josef picks an integer between 1 and 1000 inclusive and Timothy divides 1000 by that integer and states whether or not the quotient is an integer.  How many integers could Josef pick such that Timothy's quotient is an integer?

### Answer:
