AMC8 1991

AMC8 1991 · Q20

AMC8 1991 · Q20. It mainly tests Logic puzzles, Remainders & modular arithmetic.

In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then $C =$

在加法问题中，每个数字都被字母替换。如果不同的字母代表不同的数字，那么$C=$

(A) 1 1

(B) 3 3

(D) 7 7

(E) 9 9

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): For the sum to be 300, $A=2$ in the hundreds’ place, since $A=1$ gives numbers too small and $A=3,4,\ldots$ makes the sum too large. If $A=2$ then $B=7$, since $A=2$ and $B=8$ or $9$ would be too large for the ones’ place and $A=2$ and $B=6,5,\ldots$ would not be enough to carry a 1 from the tens’ to the hundreds’ place. Thus, if $A=2$ and $B=7$ and $A+B+C=10$ in the ones’ place, then $C=1$.

答案（A）：要使和为 300，百位上的 $A=2$，因为 $A=1$ 得到的数太小，而 $A=3,4,\ldots$ 会使和太大。若 $A=2$，则 $B=7$，因为当 $A=2$ 且 $B=8$ 或 $9$ 时个位会过大；而当 $A=2$ 且 $B=6,5,\ldots$ 时，又不足以从十位向百位产生进位 1。因此，若 $A=2$ 且 $B=7$，并且个位满足 $A+B+C=10$，则 $C=1$。

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