Question

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}

$2$

$4$

$7$

$8$

$9$

Solution:

From the leftmost column, there is no carry into a sixth digit, so $A+B=D\le 9$ .

The units column is $B+A=D$ , so it also has no carry. The tens column then gives $C+D=D$ , hence $C=0$ .

Since $A$ and $B$ are distinct nonzero digits, $D=A+B$ can be any digit from $3$ through $9$ . For example, $(A,B)=(1,2),(1,3),(2,3),(2,4),(2,5),(2,6),(2,7)$ give $D=3,4,5,6,7,8,9$ .

Thus there are $7$ possible values of $D$ , and the correct answer is C .

2014 AMC 10B Problem 10

Solution:

Problem 10 in Other Years