考题7-8 2017 AMC 8

AMC 8数学竞赛专为8年级及以下的初中学生设计，但近年来的数据显示，越来越多小学4-6年级的考生加入到AMC 8级别的考试行列中，而当这些学生能在成绩中取得“A”类标签，则是对孩子数学天赋的优势证明，不管是对美高申请还是今后在数学领域的发展都极其有利！那么接下来跟随小编来看一下AMC 8的官方真题以及官方解答吧：

Problem 7

Let  be a 6-digit positive integer, such as 247247, whose first three digits are the same as its last three digits taken in the same order. Which of the following numbers must also be a factor of ?

Solution 1

Let  Clearly,  is divisible by .

Solution 2

We can see that numbers like  can be written as . We can see that the alternating sum of digits is , which is . Because  is a multiple of , any number  is a multiple of , so the answer is 

-Baolan (hi MVMS)

Solution 3

The most important step is to realize that any number in the form . Thus every number in this form is divisible by , and the answer is , because it is the only choice that is a factor of .

Problem 8

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?

Solution

Notice that (1) cannot be true. Otherwise, the number would have to prime and either be even or divisible by 7. This only happens if the number is 2 or 7, neither of which are two-digit numbers, so we run into a contradiction. Thus, we must have (2), (3), and (4) true. By (2), the -digit number is even, and thus the digit in the tens place must be . The only even -digit number starting with  and divisible by  is , which has a units digit of 

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