1989 AJHSME Problems/Problem 22

Problem

The letters $\text{A}$ , $\text{J}$ , $\text{H}$ , $\text{S}$ , $\text{M}$ , $\text{E}$ and the digits $1$ , $9$ , $8$ , $9$ are "cycled" separately as follows and put together in a numbered list:

$\begin{tabular}[t]{lccc} & & AJHSME & 1989 \\ & & & \\ 1. & & JHSMEA & 9891 \\ 2. & & HSMEAJ & 8919 \\ 3. & & SMEAJH & 9198 \\ & & ........ & \end{tabular}$

What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time?

$\text{(A)}\ 6 \qquad \text{(B)}\ 10 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 24$

Solution

Every $4\text{th}$ line has $1989$ as part of it and every $6\text{th}$ line has $\text{AJHSME}$ as part of it. In order for both to be part of line $n$ , $n$ must be a multiple of $4$ and $6$ , the least of which is $\text{lcm}(4,6)=12\rightarrow \boxed{\text{C}}$ .

1989 AJHSME (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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1989 AJHSME Problems/Problem 22

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