返回题库

HMMT 二月 2003 · 冲刺赛 · 第 6 题

HMMT February 2003 — Guts Round — Problem 6

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [6] Define the Fibonacci numbers by F = 0, F = 1, F = F + F for n ≥ 2. For
    0 1 n n − 1 n − 2
    how many n , 0 ≤ n ≤ 100, is F a multiple of 13?
    HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND
    √
    √
    √

英文原题

[6] Define the Fibonacci numbers by F 0 = 0, F 1 = 1, F n = F n − 1 + F n − 2 for n ≥ 2. For
how many n , 0 ≤ n ≤ 100, is F n a multiple of 13?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HARVARD-MIT MATHEMATICS TOURNAMENT, MARCH 15, 2003 — GUTS ROUND

解析

英文解析

  1. Define the Fibonacci numbers by F = 0, F = 1, F = F + F for n ≥ 2. For
    0 1 n n − 1 n − 2
    how many n , 0 ≤ n ≤ 100, is F a multiple of 13?
    1 n
    Solution: 15
    The sequence of remainders modulo 13 begins 0 , 1 , 1 , 2 , 3 , 5 , 8 , 0, and then we have
    F ≡ 8 F modulo 13 by a straightforward induction. In particular, F is a multiplen +7 n nof 13 if and only if 7 | n , so there are 15 such n .
    √
    √
    √