2014 AMC 10B Problems/Problem 10

Problem

In the addition shown below $A$ , $B$ , $C$ , and $D$ are distinct digits. How many different values are possible for $D$ ?

$\begin{array}[t]{r} ABBCB \\ + \ BCADA \\ \hline DBDDD \end{array}$

$\textbf {(A) } 2 \qquad \textbf {(B) } 4 \qquad \textbf {(C) } 7 \qquad \textbf {(D) } 8 \qquad \textbf {(E) } 9$

Solution

Note from the addition of the last digits that $A+B=D\text{ or }D+10$ . In the latter case we must have that $1+C+D=D\text{ or }10+D$ , implying that $C=9$ . In the addition of the third digits, we then have $1+B+A=D\text{ or }D+10$ , a contradiction from our assumption that $A+B=10$ . Thus $A+B=D$ .

This then implies that $C+D=D$ , or $C=0$ . Note that all of the remaining equalities are now satisfied: $A+B=D, B+C=B,$ and $B+A=D$ . Thus, if we have some $A,B,D$ such that $A+B=D$ then the addition will be satisfied. Since the digits must be distinct, the smallest possible value of $D$ is $1+2=3$ , and the largest possible value is $9$ . Any of these values can be obtained by taking $A=1,B=D-1$ . Thus we have that $3\le D\le9$ , so the number of possible values is $\boxed{\textbf{(C) }7$ (Error compiling LaTeX. Unknown error_msg)

2014 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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All AMC 10 Problems and Solutions

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2014 AMC 10B Problems/Problem 10

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