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Numbers congruent to 1 mod k

Source: 1978 AHSME Problem 27

June 18, 2014
number theoryleast common multipleAMC

Problem Statement

There is more than one integer greater than 11 which, when divided by any integer kk such that 2≤k≤112 \le k \le 11, has a remainder of 11. What is the difference between the two smallest such integers?
<spanclass=′latex−bold′>(A)</span>2310<spanclass=′latex−bold′>(B)</span>2311<spanclass=′latex−bold′>(C)</span>27,720<spanclass=′latex−bold′>(D)</span>27,721<spanclass=′latex−bold′>(E)</span>none of these<span class='latex-bold'>(A) </span>2310\qquad<span class='latex-bold'>(B) </span>2311\qquad<span class='latex-bold'>(C) </span>27,720\qquad<span class='latex-bold'>(D) </span>27,721\qquad <span class='latex-bold'>(E) </span>\text{none of these}
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