2007 AMC 10B Problem 9

Problem 9 of 25EasierAlgebra Number Theory

A cryptographic code is designed as follows. The first time a letter appears in a given message it is replaced by the letter that is

1

place to its right in the alphabet (assuming that the letter A is one place to the right of the letter Z). The second time this same letter appears in the given message, it is replaced by the letter that is

1+2

places to the right, the third time it is replaced by the letter that is

1+2+3

places to the right, and so on. For example, with this code the word “banana” becomes “cbodqg”. What letter will replace the last letter s in the message “Lee’s sis is a Mississippi miss, Chriss!”?

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Solution

The final s is the

12

th appearance of the letter s in the message, so it is shifted

1+2+\cdots+12=\dfrac{12\cdot 13}{2}=78

places to the right. Since

78=3\cdot 26

is a multiple of the alphabet length

26,

the shift returns to the same letter, s. Thus, the correct answer is D.

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