HMMT 二月 2004 · GEN1 赛 · 第 2 题

英文解析

2. 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 atkleast d · 10 . On the other hand, each of the digits after the first is at most 9, so thekproduct of the digits is at most d · 9 . Thus, if n equals the product of its digits, thenk kd · 10 ≤ n ≤ d · 9
   which forces k = 0, i.e., the number has only one digit. So n = 9 is clearly the largestpossible value.