2000 AMC 8 Problems/Problem 9
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. 
