HMMT 二月 2004 · GEN1 赛 · 第 2 题
HMMT February 2004 — GEN1 Round — Problem 2
题目详情
- What is the largest whole number that is equal to the product of its digits?
解析
- What is the largest whole number that is equal to the product of its digits? Solution: 9 Suppose the number n has k + 1 digits, the first of which is d . Then the number is at k least d · 10 . On the other hand, each of the digits after the first is at most 9, so the k product of the digits is at most d · 9 . Thus, if n equals the product of its digits, then k k d · 10 ≤ n ≤ d · 9 which forces k = 0, i.e., the number has only one digit. So n = 9 is clearly the largest possible value.