Difference between revisions of "2000 AMC 8 Problems/Problem 9"
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| − | ==Problem== | + | == Problem == |
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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} | ||
| 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>. | ||
| − | ==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}} | ||
Revision as of 12:30, 19 October 2020
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/2/9/6/296e2ed8523d00b950d04cba19abed3767a95e7f.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(1)); label("$1$",(0,2),SE); label("$2$",(0,1),SE); [/asy]](http://latex.artofproblemsolving.com/0/f/1/0f12817a85b00f769cdccde1c3c265c31347293a.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 
.
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. 
