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

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==Problem==
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== 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}
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<math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math>
 
<math>\text{(A)}\ 0 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 8</math>
  
==Solution==
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== Solution ==
 
 
 
The <math>3</math>-digit powers of <math>5</math> are <math>125</math> and <math>625</math>, so space <math>2</math> is filled with a <math>2</math>.
 
The <math>3</math>-digit powers of <math>5</math> are <math>125</math> and <math>625</math>, so space <math>2</math> is filled with a <math>2</math>.
 
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>.
  
==See Also==
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== 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 12:30, 19 October 2020

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(1));  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 $\boxed{\text{(D) 6}}$.

See Also

2000 AMC 8 (ProblemsAnswer KeyResources)
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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