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Difference between revisions of "2000 AMC 8 Problems/Problem 9"

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== Problem ==
 
== Problem ==
−
Three-digit powers of <math>2</math> and <math>5</math> are used in this ''cross-number'' puzzle. What is the only possible digit for the outlined square?
+
Three-digit powers of <math>2</math> and <math>5</math> are used in this "cross-number" puzzle. What is the only possible digit for the outlined square?
 
<cmath>\begin{array}{lcl}
 
<cmath>\begin{array}{lcl}
 
\textbf{ACROSS} & & \textbf{DOWN} \\
 
\textbf{ACROSS} & & \textbf{DOWN} \\
−
\textbf{2}. 2^m & & \textbf{1}. 5^n
+
\textbf{2}.~ 2^m & & \textbf{1}.~ 5^n
 
\end{array}</cmath>
 
\end{array}</cmath>
  
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draw((0,-1)--(1,-1)--(1,2)--(0,2)--cycle);
 
draw((0,-1)--(1,-1)--(1,2)--(0,2)--cycle);
 
draw((0,1)--(3,1)--(3,0)--(0,0));
 
draw((0,1)--(3,1)--(3,0)--(0,0));
−
draw((3,0)--(2,0)--(2,1)--(3,1)--cycle,linewidth(1));
+
draw((3,0)--(2,0)--(2,1)--(3,1)--cycle,linewidth(2));
  
 
label("$1$",(0,2),SE);
 
label("$1$",(0,2),SE);
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The only <math>3</math>-digit power of <math>2</math> beginning with <math>2</math> is <math>256</math>, so the outlined block is filled with
 
The only <math>3</math>-digit power of <math>2</math> beginning with <math>2</math> is <math>256</math>, so the outlined block is filled with
 
a <math>\boxed{\text{(D) 6}}</math>.
 
a <math>\boxed{\text{(D) 6}}</math>.
 +
 +
==Video Solution==
 +
https://youtu.be/QAeRqTq3a7Y Soo, DRMS, NM
  
 
== See Also ==
 
== See Also ==
 
{{AMC8 box|year=2000|num-b=8|num-a=10}}
 
{{AMC8 box|year=2000|num-b=8|num-a=10}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 17:34, 28 March 2023

Problem

Three-digit powers of $2$ and $5$ are used in this "cross-number" puzzle. What is the only possible digit for the outlined square? \[\begin{array}{lcl} \textbf{ACROSS} & & \textbf{DOWN} \\ \textbf{2}.~ 2^m & & \textbf{1}.~ 5^n \end{array}\]

[asy] draw((0,-1)--(1,-1)--(1,2)--(0,2)--cycle); draw((0,1)--(3,1)--(3,0)--(0,0)); draw((3,0)--(2,0)--(2,1)--(3,1)--cycle,linewidth(2)); label("$1$",(0,2),SE); label("$2$",(0,1),SE); [/asy]

$\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8$

Solution

The $3$-digit powers of $5$ are $125$ and $625$, so space $2$ is filled with a $2$. The only $3$-digit power of $2$ beginning with $2$ is $256$, so the outlined block is filled with a $\boxed{\text{(D) 6}}$.

Video Solution

https://youtu.be/QAeRqTq3a7Y Soo, DRMS, NM

See Also

2000 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
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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