AMC12 2014 B

AMC12 2014 B · Q8

AMC12 2014 B · Q8. 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 $D$? \[ \begin{array}{r} \mathrm{ABBCB} \\ + \mathrm{BCADA} \\ \hline \mathrm{DBDDD} \end{array} \]

如下所示的加法中，$A$、$B$、$C$、$D$ 是不同的数字。$D$ 有多少个不同的可能值？ \[ \begin{array}{r} \mathrm{ABBCB} \\ + \mathrm{BCADA} \\ \hline \mathrm{DBDDD} \end{array} \]

(A) 2 2

(B) 4 4

(D) 8 8

(E) 9 9

Answer

Correct choice: (C)

正确答案：（C）

Solution

English (OCR): 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 $3$: $(1,2,0,3)$, $(1,3,0,4)$, $(2,3,0,5)$, $(2,4,0,6)$, $(2,5,0,7)$, $(2,6,0,8)$, $(2,7,0,9)$.

中文翻译：答案（C）：由最左列可知 $A+B\le 9$。因此第二列和第四列都表明 $C=0$。由于 $A$、$B$、$C$ 是互不相同的数字，$D$ 至少为 $3$。下面给出的 $(A,B,C,D)$ 的取值表明，$D$ 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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