Difference between revisions of "2006 AMC 8 Problems/Problem 24"

(Video solution)
m (Video solution)
Line 7: Line 7:
 
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 9 </math>
 
<math> \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 9 </math>
  
==Video solution==
+
==Video Solution by OmegaLearn==
 +
https://youtu.be/7an5wU9Q5hk?t=3080
 +
 
 +
==Video Solution==
 
https://youtu.be/sd4XopW76ps -Happytwin
 
https://youtu.be/sd4XopW76ps -Happytwin
 
https://youtu.be/7an5wU9Q5hk?t=3080
 
  
 
https://www.youtube.com/channel/UCUf37EvvIHugF9gPiJ1yRzQ
 
https://www.youtube.com/channel/UCUf37EvvIHugF9gPiJ1yRzQ

Revision as of 16:20, 31 December 2022

Problem

In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?

\[\begin{array}{cccc}& A & B & A\\ \times & & C & D\\ \hline C & D & C & D\\ \end{array}\]

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 9$

Video Solution by OmegaLearn

https://youtu.be/7an5wU9Q5hk?t=3080

Video Solution

https://youtu.be/sd4XopW76ps -Happytwin

https://www.youtube.com/channel/UCUf37EvvIHugF9gPiJ1yRzQ

https://www.youtube.com/watch?v=Y4DXkhYthhs

Solution

$CDCD = CD \cdot 101$, so $ABA = 101$. Therefore, $A = 1$ and $B = 0$, so $A+B=1+0=\boxed{\textbf{(A)}\ 1}$.


Solution 2

Method 1: Test $examples.$

Method 2: Bash it out to $waste$ time

$(100A+10B+A)(10C+D) = 1000C+100D+10C+D$ $1000AC+100BC+10AC+100AD+10BD+AD=1010C+101D$

$1010AC+100BC+101AD = 1010C + 101D$

$1010(A-1)(C) + 101(A-1)D + 100CB + 10BD=0$

$A=1$ ,$B=0$

And $0+1=1$, thus the answer is $A$

See Also

2006 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 23
Followed by
Problem 25
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. AMC logo.png