AMC8 2012

AMC8 2012 · Q16

AMC8 2012 · Q16. It mainly tests Rounding & estimation, Digit properties (sum of digits, divisibility tests).

Each of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 is used only once to make two five-digit numbers so that they have the largest possible sum. Which of the following could be one of the numbers?

将数字0、1、2、3、4、5、6、7、8、9各使用一次，组成两个五位数，使得它们的和尽可能大。以下哪一个可能是其中一个数？

(A) 76531 76531

(B) 86724 86724

(D) 96240 96240

(E) 97403 97403

Answer

Correct choice: (C)

正确答案：（C）

Solution

In order to maximize the sum of the numbers, the numbers must have their digits ordered in decreasing value. There are only two numbers from the answer choices with this property: $76531$ and $87431$. To determine the answer we will have to use estimation and the first two digits of the numbers. For $76531$ the number that would maximize the sum would start with $98$. The first two digits of $76531$ (when rounded) are $77$. Adding $98$ and $77$, we find that the first three digits of the sum of the two numbers would be $175$. For $87431$ the number that would maximize the sum would start with $96$. The first two digits of $87431$ (when rounded) are $87$. Adding $96$ and $87$, we find that the first three digits of the sum of the two numbers would be $183$. From the estimations, we can say that the answer to this problem is $\boxed{\textbf{(C)}\ 87431}$. p.s. USE INTUITION, see answer choices before solving any question -litttle_master

为了使两个数的和最大，两个数必须按照数字从大到小的顺序排列。选项中只有两个数具有这个性质：$76531$ 和 $87431$。为了确定答案，我们使用估算和前两位数字。对于 $76531$，与之配对使和最大的数以 $98$ 开头。$76531$ 的前两位数字（四舍五入后）是 $77$。$98 + 77 = 175$，所以两个数的和的前三位数字是 $175$。对于 $87431$，与之配对使和最大的数以 $96$ 开头。$87431$ 的前两位数字（四舍五入后）是 $87$。$96 + 87 = 183$，所以两个数的和的前三位数字是 $183$。从估算来看，本题答案是 $\boxed{\textbf{(C)}\ 87431}$。 p.s. 用直觉，先看选项再解题 -litttle_master

Topics