AMC 8 1991 Problem 20
AMC8 1991 年第 20 题
逻辑推理（跨域）★★★☆☆
In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then what is the value of ?
在下面的加法算式中，每个数字被替换为一个字母。如果不同的字母代表不同的数字，则 的值是多少？
- A.1
- B.3
- C.5
- D.7
- E.9
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
If this sum would be larger than 300, which is too large. If this sum would be at most 199 + 19 + 1 = 219, which is too small. So A = 2.
If this sum would be at least 280 + 28 + 2 = 310, which is too large. If this sum would be at most 269 + 26 + 2 = 297, which is too small. So B = 7.
This forces C = 1, for a sum of 271 + 27 + 2 = 300.
From this we have 111A+11B+C=300. Clearly, A<3. Since , Thus, A=2 and 11B+C=78. From here it becomes clear that B=7 and .
Using logic, a+b+c= 10, therefore b+a+1 (from the carry over) = 10. So b+a=9, A+1=3
Thus, A=2 and 11B+C=78. From here it becomes clear that B=7 and .
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