Question

The letters $A, J, H, S, M, E$ and the digits $1, 9, 8, 9$ are each cycled separately (shifted one place at a time) to build a numbered list. Starting from AJHSME 1989, the list begins: line $1$ is JHSMEA 9891, line $2$ is HSMEAJ 8919, line $3$ is SMEAJH 9198, and so on. On what numbered line will AJHSME 1989 appear for the first time?

$6$

$10$

$12$

$18$

$24$

Solution:

The six letters cycle back to their original order AJHSME every $6$ lines, and the four digits cycle back to $1989$ every $4$ lines.

Both happen on the same line at the least common multiple of $6$ and $4,$ which is $12.$ So AJHSME 1989 first reappears on line $12.$

Thus, the correct answer is C .

1989 AMC 8 Problem 22

Solution:

Problem 22 in Other Years