End in 2015, part 2

This is equivalent to solving for:

$995\times S \equiv 2015 \pmod{10000}$

This will give 5 solutions spaced 2000 apart for $S$ with $S\in [0,10000]$ since the gcd of 995 and 10000 is 5. We can find the solution under 2000 by solving for $S$ in

$995S+10000y=2015 \Rightarrow 199S+2000y=403$

Solve the LDE for $199S+2000y=1$ - we get by standard methods that $S=1799$ works. Hence, one possible value of $S$ is $1799 \times 403 \pmod{2000}= 997$ .

Thus the 4-digit values of $S$ are $2997,4997,6997,8997$ which have a sum of $23988$ .