2017 AMC 8 第 8 题

Question

马尔科姆今天放学后想去伊莎贝拉家。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。”

(1)

它是质数。

(2)

它是偶数。

(3)

它能被

7

整除。

(4)

它的一个数字是

9

。

这些信息足以让马尔科姆确定伊莎贝拉的门牌号。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

小提示：

恰好三个陈述为真，所以只有一个陈述为假

Since exactly three statements are true, only one statement can fail.

大提示：

一个两位数不可能既是质数又是大于

2

的偶数，也不可能既是质数又能被

7

整除。

A two-digit number cannot be both prime and an even multiple greater than

2

, and it cannot be both prime and divisible by

7

.

视频讲解：

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文字解答：

陈述

(1)

和

(2)

不能同时为真，因为唯一的偶质数不是两位数。陈述

(1)

和

(3)

也不能同时为真，因为能被

7

整除的两位数不是质数。由于只有一个陈述为假，陈述

(1)

必须为假，而陈述

(2)

、

(3)

和

(4)

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.