HMMT 十一月 2010 · 冲刺赛 · 第 8 题

英文解析

8. [ 7 ] What is the sum of all four-digit numbers that are equal to the cube of the sum of their digits
   (leading zeros are not allowed)?
   Answer: 10745 We want to find all integers x between 1000 and 9999 that are the cube of the sumof their digits. Of course, our search is only restricted to perfect cubes. The smallest such cube is
   3 3
   10 = 1000 and the largest such cube is 21 = 9261. This means we only have to check 12 differentcubes, which is quite doable, but we can reduce the search even further with a little number theory.
   Suppose we write our number as x = 1000 a + 100 b + 10 c + d , where a , b , c , and d are the decimal digitsof x . Then we have
   ( a + b + c + d ) ≡ 1000 a + 100 b + 10 c + d ≡ a + b + c + d (mod 9)3
   If we let k = a + b + c + d , then k must be a solution to the modular equation k ≡ k (mod 9). A3
   quick check of the values 0 through 8 shows that the only solutions are 0, 1, and 8.
   Now, in our search, we only have to check values that are the cube of a number which is either 0, 1, or
   8 mod 9.
   10 = 1000, but 1 + 0 + 0 + 0 6 = 10.3
   17 = 4913, and 4 + 9 + 1 + 3 = 17.3
   18 = 5832, and 5 + 8 + 3 + 2 = 18.3
   19 = 6859, but 6 + 8 + 5 + 9 6 = 19.3
   So the only solutions are 4913 and 5832, which sum to 10745.
   Guts Round