Which one of the following integers can be expressed as the sum of $100$ consecutive positive integers?
$\textbf{(A)}\ 1,\!627,\!384,\!950\qquad\textbf{(B)}\ 2,\!345,\!678,\!910\qquad\textbf{(C)}\ 3,\!579,\!111,\!300\qquad\textbf{(D)}\ 4,\!692,\!581,\!470\qquad\textbf{(E)}\ 5,\!815,\!937,\!260$
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 $\boxed{A}$, 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 $16273899$.