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Integers dividing large numbers

Source: 1970 AHSME Problem 34

April 16, 2014
modular arithmeticgreatest common divisorAMC

Problem Statement

The greatest integer that will divide 13,51113,511, 13,90313,903, and 14,58914,589 and leave the same remainder is
<spanclass=′latex−bold′>(A)</span>28<spanclass=′latex−bold′>(B)</span>49<spanclass=′latex−bold′>(C)</span>98<span class='latex-bold'>(A) </span>28\qquad<span class='latex-bold'>(B) </span>49\qquad<span class='latex-bold'>(C) </span>98\qquad
<spanclass=′latex−bold′>(D)</span>an odd multiple of 7 greater than 49<spanclass=′latex−bold′>(E)</span>an even multiple of 7 greater than 98<span class='latex-bold'>(D) </span>\text{an odd multiple of }7\text{ greater than }49\qquad <span class='latex-bold'>(E) </span>\text{an even multiple of }7\text{ greater than }98
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