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 | + | <cmath>\begin{array}{lcl} |
| − | <cmath>\begin{ | ||
\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{ | + | \end{array}</cmath> |
<asy> | <asy> | ||
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( | + | draw((3,0)--(2,0)--(2,1)--(3,1)--cycle,linewidth(2)); |
label("$1$",(0,2),SE); | label("$1$",(0,2),SE); | ||
| Line 18: | Line 17: | ||
<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== | + | == 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>. | ||
| − | == | + | ==Video Solution== |
| + | https://youtu.be/QAeRqTq3a7Y Soo, DRMS, NM | ||
| + | == See Also == | ||
{{AMC8 box|year=2000|num-b=8|num-a=10}} | {{AMC8 box|year=2000|num-b=8|num-a=10}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 17:34, 28 March 2023
Contents
Problem
Three-digit powers of 
and 
are used in this "cross-number" puzzle. What is the only possible digit for the outlined square?
![\[\begin{array}{lcl} \textbf{ACROSS} & & \textbf{DOWN} \\ \textbf{2}.~ 2^m & & \textbf{1}.~ 5^n \end{array}\]](http://latex.artofproblemsolving.com/a/8/7/a877d37c9a04e625cb9fdc9f59c1f3ddc78746bb.png)
![[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]](http://latex.artofproblemsolving.com/2/7/3/273126e87eac499d2e6ba1b00aed3bd4b97258d7.png)
Solution
The 
-digit powers of are 
and 
, so space is filled with a .
The only -digit power of beginning with is 
, so the outlined block is filled with
a 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
