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Difference between revisions of "1989 AJHSME Problems/Problem 22"

(New page: ==Problem== The letters <math>\text{A}</math>, <math>\text{J}</math>, <math>\text{H}</math>, <math>\text{S}</math>, <math>\text{M}</math>, <math>\text{E}</math> and the digits <math>1</ma...)
 
 
Line 2: Line 2:
  
 
The letters <math>\text{A}</math>, <math>\text{J}</math>, <math>\text{H}</math>, <math>\text{S}</math>, <math>\text{M}</math>, <math>\text{E}</math> and the digits <math>1</math>, <math>9</math>, <math>8</math>, <math>9</math> are "cycled" separately as follows and put together in a numbered list:
 
The letters <math>\text{A}</math>, <math>\text{J}</math>, <math>\text{H}</math>, <math>\text{S}</math>, <math>\text{M}</math>, <math>\text{E}</math> and the digits <math>1</math>, <math>9</math>, <math>8</math>, <math>9</math> are "cycled" separately as follows and put together in a numbered list:
 +
 
<cmath>\begin{tabular}[t]{lccc}
 
<cmath>\begin{tabular}[t]{lccc}
 
  & & AJHSME & 1989 \\
 
  & & AJHSME & 1989 \\

Latest revision as of 16:03, 19 April 2021

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 $\text{lcm}(4,6)=12\rightarrow \boxed{\text{C}}$.

See Also

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
All AJHSME/AMC 8 Problems and Solutions
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