HMMT 十一月 2010 · 冲刺赛 · 第 10 题

英文解析

10. [ 8 ] What is the smallest integer greater than 10 such that the sum of the digits in its base 17 representation is equal to the sum of the digits in its base 10 representation?
    Answer: 153 We assume that the answer is at most three digits (in base 10). Then our desirednumber can be expressed in the form abc = def , where a, b, c are digits in base 10, and d, e, f are
    1710
    digits in base 17. These variables then satisfy the equations
    100 a + 10 b + c = 289 d + 17 e + f,
    a + b + c = d + e + f.
    Subtracting the second equation from the first, we obtain 99 a + 9 b = 288 d + 16 e , or 9(11 a + b ) =
    16(18 d + e ). From this equation, we find that 11 a + b must be divisible by 16, and 18 d + e must bedivisible by 9. To minimize abc , we find the minimal possible value of a : If a = 0, then the only way for
    11 a + b = b to be divisible by 16 is to set b = 0; however, this is disallowed by the problem condition,
    which stipulates that the number must be greater than 10. If we try a = 1, then we find that theonly possible value of b which lets 11 a + b = b + 11 be divisible by 16 is b = 5. Plugging these in andsimplifying, we find that we must have 18 d + e = 9. The only possible solution to this is d = 0 , e = 9.
    Now to satisfy a + b + c = d + e + f , we must have 1 + 5 + c = 0 + 9 + f , or c = f + 3. The minimalpossible solution to this is c = 3 , f = 0. So our answer is abc = 153 , which is also equal to 090 .
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