AMC8 2017

AMC8 2017 · Q8

AMC8 2017 · Q8. It mainly tests Logic puzzles, Divisibility & factors.

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?

马尔科姆放学后想去拜访伊莎贝拉，他知道她住的街道但不知道她的门牌号。她告诉他：“我的门牌号是两位数，且以下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) 它有一个数字是 9。这个信息让马尔科姆能确定伊莎贝拉的门牌号。它的个位数字是多少？

(A) 4 4

(B) 6 6

(D) 8 8

(E) 9 9

Answer

Correct choice: (D)

正确答案：（D）

Solution

Answer (D): There are no two-digit even primes, so statements (1) and (2) cannot both be true. Also, no two-digit prime is divisible by 7, so statements (1) and (3) cannot both be true. Because there is only one false statement, it must be (1), so Isabella’s house number is an even two-digit multiple of 7 that has a digit of 9. The number is even, so the 9 must be the tens digit. The only even multiple of 7 between 90 and 99 is 98, so the units digit is 8.

答案 (D)：没有两位的偶质数，所以陈述 (1) 和 (2) 不可能同时为真。另外，没有两位质数能被 7 整除，所以陈述 (1) 和 (3) 也不可能同时为真。由于只有一个陈述是假的，那必定是 (1)，因此 Isabella 的门牌号是一个含有数字 9 的两位偶数且是 7 的倍数。该数是偶数，所以 9 必然在十位。90 到 99 之间唯一的 7 的偶倍数是 98，所以个位数字是 8。

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