HMMT 二月 2015 · 冲刺赛 · 第 17 题

英文解析

17. [ 11 ] Find the least positive integer N > 1 satisfying the following two properties:
    • There exists a positive integer a such that N = a (2 a − 1).
    • The sum 1 + 2 + · · · + ( N − 1) is divisible by k for every integer 1 ≤ k ≤ 10.
    Answer: 2016 The second condition implies that 16 divides a (2 a − 1)(2 a − a − 1), which shows 2
    that a ≡ 0 or 1 modulo 16. The case a = 1 would contradict the triviality-avoiding condition N > 1.
    a cannot be 16, because 7 does not divide a (2 a − 1)(2 a − a − 1). a cannot be 17, because 9 does not 2
    divide a (2 a − 1)(2 a − a − 1). It can be directly verified that a = 32 is the smallest positive integer 2
    4 2
    for which 1 + 2 + · · · + ( N − 1) = 2 · 3 · 5 · 7 · 13 · 31 which is divisible by 1 , 2 , . . . , 10. For this a , wecompute N = 32(2 · 32 − 1) = 2016.