AMC12 2017 A

AMC12 2017 A · Q12

AMC12 2017 A · Q12. It mainly tests GCD & LCM, Remainders & modular arithmetic.

There are 10 horses, named Horse 1, Horse 2, \dots, 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，名为 Horse 1, Horse 2, \dots, Horse 10。它们的名字来自跑完一个圆形赛道的圈所需的时间：Horse $k$ 跑一圈正好需要 $k$ 分钟。在时间 0，所有马都在赛道起点一起。马开始朝同一方向跑，并以恒定速度在圆形赛道上持续跑。最小的 $S>0$ 时间（分钟），10 匹马再次同时回到起点是 $S=2520$。让 $T>0$ 为最小的分钟数，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 $k$ divides $t$. Let $I(t)$ be the number of integers $k$ with $1\le k\le 10$ that divide $t$. Then $I(1)=1$, $I(2)=2$, $I(3)=2$, $I(4)=3$, $I(5)=2$, $I(6)=4$, $I(7)=2$, $I(8)=4$, $I(9)=3$, $I(10)=4$, $I(11)=1$, $I(12)=5$. Thus $T=12$ and the 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 $t$。让 $I(t)$ 为 $1\le k\le 10$ 中整除 $t$ 的整数个数。那么 $I(1)=1$，$I(2)=2$，$I(3)=2$，$I(4)=3$，$I(5)=2$，$I(6)=4$，$I(7)=2$，$I(8)=4$，$I(9)=3$，$I(10)=4$，$I(11)=1$，$I(12)=5$。因此 $T=12$，各位数字和为 $1+2=3$。

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