HMMT 十一月 2010 · 冲刺赛 · 第 10 题

HMMT November 2010 — Guts Round — Problem 10

[ 8 ] What is the smallest integer greater than 10 such that the sum of the digits in its base 17 repre- sentation is equal to the sum of the digits in its base 10 representation?

解析

[ 8 ] What is the smallest integer greater than 10 such that the sum of the digits in its base 17 repre- sentation 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 desired number can be expressed in the form abc = def , where a, b, c are digits in base 10, and d, e, f are 10 17 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 be divisible 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 the only possible value of b which lets 11 a + b = b + 11 be divisible by 16 is b = 5. Plugging these in and simplifying, 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 minimal possible solution to this is c = 3 , f = 0. So our answer is abc = 153 , which is also equal to 090 . 17