smc

The sum of the first $100$ integers is $\frac{100\cdot 101}{2}=5050$. If you add an integer $k$ to each of the $100$ numbers, you get $5050+100k$, which is the sum of the numbers from $k+1$ to $k+100$. You're only adding multiples of $100$, so the last two digits will remain unchanged. Thus, the only possible answer is $\boxed{1,627,384,950}$, because the last two digits are $50$. As an aside, if $5050+100k=1627384950$, then $k=16273799$, and the numbers added are the integers from $16273800$ to $16273899$.