An then

$A_n=\dfrac{5n}{4}$ This is an integer when $4|n$ (4 divides $n$ ). There are $\boxed{251}$ multiples of 4 such that $0\le n\le 1000$ .

n must be divisible by 4 (because only $4\cdot1.25$ is an integer so $\frac{1000}{4}=250$ Then, we have to add 1 to account for 0, because n can be 0 and the 250 numbers are positive multiples of 4. Therefore, our answer is: $250+1=\boxed{251}$

We can write 1.25n as 5n/4. 5n/4 is integer only when n is a multiple of 4. Between 0 and 1000 there are 250 multiples of 4. ((1000/4)=250). And also 0 is multiple of every number. So, total 251 possibilties of n.

1.25n will only ever be an integer when n > 0 is a multiple of 4, so will only be an integer for quarter of the integers from 1 - 1000, giving us 250.

We can also count n = 0 as an answer, giving us 251 values of n, for which 1.25n is an integer..

We can write $1.25n=(1+\frac{1}{4})n=(\frac{5}{4})n=\frac{5n}{4}$ . It is an integer iff $4\mid 5n$ . But $4\nmid 5$ , so we need $4\mid n$ . This happens when $n\in \{0\times 4, 1\times 4, ..., 250\times 4$ , a total of $\boxed{251}$ possible values.

Because $1n$ is always an integer we disregard that and focus on the $0.25n$ . We see that $0.25n$ is an integer every 4th number because $\frac{1}{0.25} = 4$ . $\frac{1000}{4} = 250$ . We now have $250$ values and also add 1 for $n = 0$ , the answer is $\boxed{251}$