HMMT 二月 2002 · 冲刺赛 · 第 55 题

英文解析

55. A sequence of positive integers is given by a = 1 and a = gcd( a , n ) + 1 for
    1 n n − 1
    n > 1. Calculate a .
    2002
    Solution: 3 . It is readily seen by induction that a ≤ n for all n . On the other hand,
    a is one greater than a divisor of 1999. Since 1999 is prime, we have a = 2 or 2000;n
    1999 1999
    the latter is not possible since 2000 > 1999, so we have a = 2. Now we straightforwardly
    1999
    compute a = 3 , a = 4, and a = 3.
    2000 2001 2002 2