2014 AMC 10B Problems/Problem 10

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Problem

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

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

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)

See Also

2014 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
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All AMC 10 Problems and Solutions

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