AMC10 2017 A

AMC10 2017 A · Q16

AMC10 2017 A · Q16. It mainly tests GCD & LCM, Counting divisors.

There are 10 horses, named Horse 1, Horse 2, … , Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse k runs one lap in exactly k minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time \(S > 0\), in minutes, at which all 10 horses will again simultaneously be at the starting point is \(S = 2520\). Let \(T > 0\) be the least time, in minutes, such that at least 5 of the horses are again at the starting point. What is the sum of the digits of T?

有10匹马，编号为Horse 1, Horse 2, … , Horse 10。它们的名称来源于它们跑完一个圆形赛道的单圈所需的时间：Horse k 跑一圈正好需要k分钟。在时间0，所有马都在赛道的起点上。马匹开始朝同一方向以恒定速度在圆形赛道上跑步。它们再次同时回到起点的最近时间\(S > 0\)，单位分钟，是\(S = 2520\)。让\(T > 0\)是这样的最小时间，使得至少5匹马再次回到起点。T的各位数字之和是多少？

(A) 2 2

(B) 3 3

(D) 5 5

(E) 6 6

Answer

Correct choice: (B)

正确答案：（B）

Solution

Horse \(k\) will again be at the starting point after \(t\) minutes if and only if \(k\) is a divisor of \(t\). Let \(I(t)\) be the number of integers \(k\) with \(1 \le k \le 10\) that divide \(t\). Then \(I(1) = 1\), \(I(2) = 2\), …, \(I(12) = 5\). Thus \(T = 12\) and the requested sum of digits is \(1 + 2 = 3\).

Horse \(k\) 在t分钟后再次回到起点当且仅当\(k\) 整除\(t\)。令\(I(t)\) 为满足\(1 \le k \le 10\)且\(k\) 整除\(t\) 的整数\(k\) 的个数。那么\(I(1) = 1\)，\(I(2) = 2\)，…，\(I(12) = 5\)。因此\(T = 12\)，要求的各位数字之和是\(1 + 2 = 3\)。

Topics