HMMT 二月 2003 · 冲刺赛 · 第 14 题

英文解析

14. A positive integer will be called “sparkly” if its smallest (positive) divisor, other than
    1, equals the total number of divisors (including 1). How many of the numbers
    2 , 3 , . . . , 2003 are sparkly?
    Solution: 3
    Suppose n is sparkly; then its smallest divisor other than 1 is some prime p . Hence,
    e ee
    1 2
    n has p divisors. However, if the full prime factorization of n is p p · · · p , ther
    1 2
    number of divisors is ( e + 1)( e + 1) · · · ( e + 1). For this to equal p , only one factorr
    1 2 rcan be greater than 1, so n has only one prime divisor — namely p — and we getp − 1 p − 1
    e = p − 1 ⇒ n = p . Conversely, any number of the form p is sparkly. There are
    1 2 41
    just three such numbers in the desired range (2 , 3 , 5 ), so the answer is 3.