HMMT 二月 2005 · 冲刺赛 · 第 20 题

英文解析

20. If n is a positive integer, let s ( n ) denote the sum of the digits of n . We say that nis zesty if there exist positive integers x and y greater than 1 such that xy = n ands ( 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 eitherx is a two-digit number or y is. Assume without loss of generality that it is x . Ifx = 10 a + b , 1 ≤ a ≤ 9 and 0 ≤ b ≤ 9, then n = 10 ay + by . If both ay and by areless than 10, then s ( n ) = ay + by , but if either is at least 10, then s ( n ) < ay + by . Itfollows that the two digits of n share a common factor greater than 1, namely y . It isnow easy to count the zesty two-digit numbers by first digit starting with 2; there area total of 5 + 4 + 5 + 2 + 7 + 2 + 5 + 4 = 34.
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