2006 AMC 8 Problems/Problem 24

Revision as of 00:07, 6 June 2021 by Mathmagicops (talk | contribs) (Solution)

Problem

In the multiplication problem below $A$, $B$, $C$, $D$ and 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

https://youtu.be/sd4XopW76ps -Happytwin

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

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

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: When you do bash:

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

And now 0+1=2. I mean, 1. So the answer is \boxed{\textbf{(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

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