1978 AHSME Problem 27

There is more than one integer greater than 1 which, when divided by any integer k such that 2 \le k \le 11, has a remainder of 1. What is the difference between the two smallest such integers?

\textbf{(A) }2310\qquad \textbf{(B) }2311\qquad \textbf{(C) }27,720\qquad \textbf{(D) }27,721\qquad \textbf{(E) }\text{none of these}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.