AMC10 2014 B

AMC10 2014 B · Q10

AMC10 2014 B · Q10. It mainly tests Logic puzzles, Digit properties (sum of digits, divisibility tests).

In the addition shown below A, B, C, and D are distinct digits. How many different values are possible for D? $$ \begin{array}{r|r} ABBCB \\ + BCADA \\ \hline DBDDD \end{array} $$

如下加法所示，A、B、C、D 是不同的数字。D 有多少种不同的可能值？ $$ \begin{array}{r|r} ABBCB \\ + BCADA \\ \hline DBDDD \end{array} $$

(A) 2 2

(B) 4 4

(D) 8 8

(E) 9 9

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): As indicated by the leftmost column $A+B\le 9$. Then both the second and fourth columns show that $C=0$. Because $A$, $B$, and $C$ are distinct digits, $D$ must be at least 3. The following values for $(A,B,C,D)$ show that $D$ may be any of the 7 digits that are at least 3: $(1,2,0,3)$, $(1,3,0,4)$, $(2,3,0,5)$, $(2,4,0,6)$, $(2,5,0,7)$, $(2,6,0,8)$, $(2,7,0,9)$.

答案（C）：如最左列所示，$A+B\le 9$。然后第二列和第四列都表明$C=0$。由于$A$、$B$和$C$是互不相同的数字，$D$必须至少为3。下面给出的$(A,B,C,D)$取值表明，$D$可以是任意一个不小于3的7个数字之一：$(1,2,0,3)$、$(1,3,0,4)$、$(2,3,0,5)$、$(2,4,0,6)$、$(2,5,0,7)$、$(2,6,0,8)$、$(2,7,0,9)$。

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