HMMT 十一月 2012 · THM 赛 · 第 5 题
HMMT November 2012 — THM Round — Problem 5
题目详情
英文原题
[7] Given any positive integer, we can write the integer in base 12 and add to gether the digits of its
base 12 representation. We perform this operation on the number 7654321
repeatedly until a single base
12 digit remains. Find this digit.
Rock-paper-scissors
In the game of rock-paper-scissors, two players each choose one of rock, pa per, or scissors to play.
Rock beats scissors, scissors beats paper, and paper beats rock. If the pla yers play the same thing, the
match is considered a draw.
解析
英文解析
- [ 7 ] Given any positive integer, we can write the integer in base 12 and add together the digits of its
4321
base 12 representation. We perform this operation on the number 7 repeatedly until a single base 65
12 digit remains. Find this digit.
Answer: 4 For a positive integer n , let s ( n ) be the sum of digits when n is expressed in base 12. Wek k − 1
claim that s ( n ) ≡ n (mod 11) for all positive integers n . Indeed, if n = d 12 + d 12 + · · · + dk k − 1 0
with each d an integer between 0 and 11, inclusive, because 12 ≡ 1 (mod 11), reducing modulo 11
1 i
5432
gives exactly s ( n ). Thus, our answer is congruent to N = 7 modulo 11, and furthermore must be 6
a one-digit integer in base 12; these two conditions uniquely determine the answer.
4321
10 5
By Fermat’s Little Theorem, 7 ≡ 1 (mod 11), and also observe that 6 ≡ 6 (mod 10) because
6 2 2
∼ ∼
6 N ≡ 7 ≡ 343 ≡ 2 ≡ 4 (mod 11), which is our answer.
= 0 (mod 2) and 6 = 1 (mod 5). Thus,
(Additionally, we note that this process of writing the number in base twelve and summing the digitsmust eventually terminate because the value decreases after each step.)
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