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

Revision as of 01:07, 6 June 2021

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

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

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

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

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)}

2006 AMC 8 (Problems • Answer Key • Resources)
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

@@ Line 17: / Line 17: @@
 <math>CDCD = CD \cdot 101</math>, so <math>ABA = 101</math>. Therefore, <math>A = 1</math> and <math>B = 0</math>, so <math>A+B=1+0=\boxed{\textbf{(A)}\ 1}</math>.
+==Solution 2==
+Method 1: Test examples
+Method 2: When you do bash:
+<math>(100A+10B+A)(10C+D) = 100C+100D+10C+D</math>
+<math>100AC+100BC+100AC+100AD+10BD+AD=1010C+101D</math>
+<math>1010(A-1)(C) + 101(A-1)D + 100CB+10D=0</math>
+<math>A=1</math>
+<math>D=0</math>
+<math>B=0</math>
+And now 0+1=2. I mean, 1. So the answer is \boxed{\textbf{(A)}
 ==See Also==
 {{AMC8 box|year=2006|n=II|num-b=23|num-a=25}}
 {{MAA Notice}}