HMMT 二月 2005 · 冲刺赛 · 第 20 题

HMMT February 2005 — Guts Round — Problem 20

[8] If n is a positive integer, let s ( n ) denote the sum of the digits of n . We say that n is zesty if there exist positive integers x and y greater than 1 such that xy = n and s ( x ) s ( y ) = s ( n ). How many zesty two-digit numbers are there? ◦

解析

If n is a positive integer, let s ( n ) denote the sum of the digits of n . We say that n is zesty if there exist positive integers x and y greater than 1 such that xy = n and s ( x ) s ( y ) = s ( n ). How many zesty two-digit numbers are there? Solution: 34 Let n be a zesty two-digit number, and let x and y be as in the problem statement. Clearly if both x and y are one-digit numbers, then s ( x ) s ( y ) = n 6 = s ( n ). Thus either x is a two-digit number or y is. Assume without loss of generality that it is x . If x = 10 a + b , 1 ≤ a ≤ 9 and 0 ≤ b ≤ 9, then n = 10 ay + by . If both ay and by are less than 10, then s ( n ) = ay + by , but if either is at least 10, then s ( n ) < ay + by . It follows that the two digits of n share a common factor greater than 1, namely y . It is now easy to count the zesty two-digit numbers by first digit starting with 2; there are a total of 5 + 4 + 5 + 2 + 7 + 2 + 5 + 4 = 34. ◦