# Difference between revisions of "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 $\text{lcm}(4,6)=12\rightarrow \boxed{\text{C}}$.