2014 AMC 10B Problem 10

Question

In the addition shown below

A,

B,

C,

and

D

are distinct digits. How many different values are possible for

D?

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

2

4

7

8

9

Column addition forces

C=0

After that, identify which digits can appear as

D=A+B

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 .

Problem 10 in other years