Question

Malcolm 今天放学后想去 Isabella 家，他知道她住在哪条街，但不知道门牌号。她告诉他：“我的门牌号是两位数，并且下列四个陈述中恰好有三个是真的。”

(一) 它是质数。

(二) 它是偶数。

(三) 它能被七整除。

(四) 它的一个数字是九。

这些信息使 Malcolm 能确定 Isabella 的门牌号。它的个位数字是多少？

Malcolm wants to visit Isabella after school today and knows the street where she lives but doesn't know her house number. She tells him, "My house number has two digits, and exactly three of the following four statements about it are true."

(1) It is prime.

(2) It is even.

(3) It is divisible by 7.

(4) One of its digits is 9.

This information allows Malcolm to determine Isabella's house number. What is its units digit?

$4$

$6$

$7$

$8$

$9$

答案：D

知识点：逻辑推理整除性

难度评级：1240

视频讲解：

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文字解答：

陈述 (一) 和 (二) 不能同时为真，因为唯一的偶质数不是两位数。陈述 (一) 和 (三) 也不能同时为真，因为两位且能被 $7$ 整除的数不是质数。由于只有一个陈述为假，陈述 (一) 必须为假，而 (二)、(三)、(四) 为真。

门牌号能被 $2$ 和 $7$ 整除，所以能被 $14$ 整除。在两位的 $14$ 的倍数中，唯一含有数字 $9$ 的是 $98$ 。因此个位数字是 $8$ 。

所以正确答案是 D。

Statements (1) and (2) cannot both be true, because the only even prime is not two-digit. Statements (1) and (3) also cannot both be true, because a two-digit number divisible by $7$ would not be prime. Since only one statement is false, statement (1) must be false, while statements (2), (3), and (4) are true.

The house number is divisible by $2$ and $7,$ so it is divisible by $14.$ Among the two-digit multiples of $14,$ the only one with a digit of $9$ is $98.$ Therefore, the units digit is $8.$

Thus, D is the correct answer.

2017 AMC 8 第 8 题

答案：D

难度评级：1240

视频讲解：

文字解答：

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