Question

设 $N$ 是数字乘积为 $120$ 的最大五位数。 $N$ 的各位数字之和是多少？

Let $N$ be the greatest five-digit number whose digits have a product of $120.$ What is the sum of the digits of $N?$

$15$

$16$

$17$

$18$

$20$

答案：D

知识点：数字最优化

难度评级：1140

视频讲解：

Click to load, then click again to play

文字解答：

为了使 $5$ 位数最大，必须先让第一位，也就是万位，尽可能大。

小于 $10$ 且能整除 $120$ 的最大数字是 $8$ ，所以第一位必须是 $8$ 。剩下各位数字的乘积为 $15$ 。

同样地，现在要让第二位尽可能大。

小于 $10$ 且能整除 $15$ 的最大数字是 $5$ ，所以第二位是 $5$ 。剩下各位数字的乘积为 $3$ 。

接着让第三位尽可能大。

小于 $10$ 且能整除 $3$ 的最大数字是 $3$ ，所以第三位是 $3$ 。剩下的乘积是 $1$ ，这说明第四位和第五位都是 $1$ 。

于是 $N = 85311$ ，各位数字之和为 $8+5+3+1+1=18$

所以正确答案是 D。

To make the largest possible $5$ digit number, we must maximize the first digit (the digit in the ten-thousands place).

The largest number that is strictly less than $10$ and divides $120$ is $8,$ so the first digit must be $8.$ Therefore, the product of the remaining number is $15.$

Similarly, we must now maximize the second digit.

The largest number that is less than $10$ and divides $15$ is $5,$ so the second digit is $5.$ Therefore, the product of the remaining number is $3.$

We must then maximize the third digit.

The largest number that is less than $10$ and divides $3$ is $3,$ so the third digit is $3.$ Therefore, the product of the remaining number is $1.$ This means the 4th and 5th digits are $1.$

This makes $N = 85311,$ so the sum of the digits is $8+5+3+1+1=18$

Thus, D is the correct answer.

2018 AMC 8 第 14 题

答案：D

难度评级：1140

视频讲解：

文字解答：

其他年份的第 14 题