AMC 8 2013 Problem 13
AMC8 2013 年第 13 题
余数与同余★★★☆☆
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
当 Clara 把她的所有的考试分数相加时，她不小心把某个分数的十位数字和个位数字调换了。那么她不正确的总分和正确总分可能相差多少？
- A.45
- B.46
- C.47
- D.48
- E.49
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
Switching a digit from the units column to the tens column increases the sum by 9 9 times the value of that digit. For example, switching a 7 7 from the units column to the tens column increases the sum by 70 − 7 = 63 = 9 ⋅ 7 70−7=63=9⋅7. Similarly, switching a digit from the tens column to the units column decreases the sum by 9 9 times the value of that digit. Therefore reversing two digits changes the sum by an amount that must be a multiple of 9 9. Among the given choices, only 45 45 is a possible difference. (Note: Other multiples of 9 9 are also possible.) Answer: 𝐴 A .
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