Question

有 $10$ 匹马，名字分别是 Horse $1$ 、Horse $2$ 、……、Horse $10$ 。Horse $k$ 跑完圆形赛道一圈恰好需要 $k$ 分钟。时刻 $0$ ，所有马都在起点。它们沿同一方向以恒定速度奔跑。

所有 $10$ 匹马再次同时回到起点的最小正时间 $S > 0$ 为 $S=2520$ 分钟。设 $T > 0$ 是至少 $5$ 匹马再次同时在起点的最小正时间。 $T$ 的各位数字之和是多少？

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 $T?$

$2$

$3$

$4$

$5$

$6$

答案：B

知识点：整除性最小公倍数

难度评级：1370

解答：

Horse $k$ 在 $t$ 分钟后回到起点，当且仅当 $k\mid t$ 。因此要找最小正整数 $t$ ，使它能被 $1,2,\ldots,10$ 中至少五个数整除。

小于 $12$ 的数都不行：例如 $6$ 只被 $1,2,3,6$ 整除， $8$ 只被 $1,2,4,8$ 整除， $9$ 只被 $1,3,9$ 整除， $10$ 只被 $1,2,5,10$ 整除。

$12$ 能被 $1,2,3,4$ 和 $6$ 整除，所以 $T=12$ ，各位数字之和为 $1+2=3$ 。

所以正确答案是 B。

Horse $k$ is back at the starting point after $t$ minutes exactly when $k\mid t$ . Thus we need the least positive $t$ that is divisible by at least five of the integers $1,2,\ldots,10$ .

Checking upward, no number below $12$ has five divisors from this list: for example, $6$ has $1,2,3,6$ , $8$ has $1,2,4,8$ , $9$ has $1,3,9$ , and $10$ has $1,2,5,10$ .

The number $12$ is divisible by $1,2,3,4,$ and $6$ , so the least possible time is $T=12$ . The sum of its digits is $1+2=3$ .

Thus, B is the correct answer.

2017 AMC 10A 第 16 题

答案：B

难度评级：1370

解答：

其他年份的第 16 题