How many integers between 1000 and 2000 have all three of the numbers 15, 20 and 25 as factors?
A number with 15, 20 and 25 as factors must be divisible by their least common multiple (LCM).  Because $15 = 3
\times 5$, $20 = 2^2 \times 5$, and $25 = 5^2$, the LCM of 15, 20 and 25 is $2^2 \times 3 \times 5^2 = 300$. There are $\boxed{3}$ multiples of 300 between 1000 and 2000: 1200, 1500 and 1800.