jmc

We have $\gcd (n,100)=\mathrm{lcm}[n,100]-450$. Since $\mathrm{lcm}[n,100]$ is a multiple of $100$, we infer that $\gcd (n,100)$ is a multiple of $50$ but not of $100$. But $\gcd (n,100)$ is also a divisor of $100$, so it can only be $50$.

This implies two conclusions: first, $n$ is a multiple of $50$ (but not of $100$); second, $$\mathrm{lcm}[n,100]=\gcd (n,100)+450=50+450=500.$$In particular, $n$ is less than $500$, so we need only check the possibilities $n=50,150,250,350,450$. Of these, only $250$ satisfies our second conclusion, so $n=250$ is the unique solution -- and the sum of all solutions is thus $\boxed{250}$.