Question

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 $T?$

$2$

$3$

$4$

$5$

$6$

Solution:

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 Problem 16

Solution:

Problem 16 in Other Years