AMC8 2013

AMC8 2013 · Q13

AMC8 2013 · Q13. It mainly tests Divisibility & factors, Digit properties (sum of digits, divisibility tests).

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计算她的总分时，她不小心将一个分数的个位数和十位数颠倒了。她的错误总分与正确总分可能相差以下哪一项？

(A) 45 45

(B) 46 46

(D) 48 48

(E) 49 49

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let the two digits be $a$ and $b$. The correct score was $10a+b$. Clara misinterpreted it as $10b+a$. The difference between the two is $|9a-9b|$ which factors into $|9(a-b)|$. Therefore, since the difference is a multiple of 9, the only answer choice that is a multiple of 9 is $\boxed{\textbf{(A)}\ 45}$.

令两个数字为 $a$ 和 $b$。正确分数为 $10a+b$。Clara误认为是 $10b+a$。两者的差为 $|9a-9b|$，可因式分解为 $|9(a-b)|$。因此，差是9的倍数，选项中唯一是9的倍数的是 $\boxed{\textbf{(A)}\ 45}$。

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