MathDB
1987 AMC 12 #16 - Cryptography and Bases

Source:

December 31, 2011
AMC

Problem Statement

A cryptographer devises the following method for encoding positive integers. First, the integer is expressed in base 55. Second, a 1-to-1 correspondence is established between the digits that appear in the expressions in base 55 and the elements of the set {V,W,X,Y,Z}\{V, W, X, Y, Z\}. Using this correspondence, the cryptographer finds that three consecutive integers in increasing order are coded as VYZVYZ, VYXVYX, VVWVVW, respectively. What is the base-10 expression for the integer coded as XYZXYZ?
<spanclass=latexbold>(A)</span> 48<spanclass=latexbold>(B)</span> 71<spanclass=latexbold>(C)</span> 82<spanclass=latexbold>(D)</span> 108<spanclass=latexbold>(E)</span> 113 <span class='latex-bold'>(A)</span>\ 48 \qquad<span class='latex-bold'>(B)</span>\ 71 \qquad<span class='latex-bold'>(C)</span>\ 82 \qquad<span class='latex-bold'>(D)</span>\ 108 \qquad<span class='latex-bold'>(E)</span>\ 113