What is the multiplicative inverse of 28 modulo 1000?

To find the multiplicative inverse of 28 modulo 1000, we solve for $x$ in the equation:

$$28x\equiv 1(\mathrm{mod}1000)$$

We use the Extended Euclidean Algorithm to find integers $x$ and $y$ such that:

$$28x+1000y=1$$

Applying the algorithm:

1. $1000=35\times 28+20$
2. $28=1\times 20+8$
3. $20=2\times 8+4$
4. $8=2\times 4+0$

The GCD is 4, which does not divide 1. Thus, no inverse exists for 28 modulo 1000 because 28 and 1000 are not coprime (their GCD is 4 ≠ 1).

Answer: $\boxed{\text{No multiplicative inverse exists.}}$