Question

There are $10$ horses, named Horse $1,$ Horse $2,$ $\ldots,$ 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 $S\gt0,$ in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S=2520.$ Let $T\gt0$ 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 at the starting point at time $t$ precisely when $k\mid t.$ So we want the smallest $t$ with at least $5$ divisors among $1,2,\ldots,10.$

Checking small values, $t=12$ is divisible by $1,2,3,4,$ and $6,$ giving exactly $5$ such horses, and no smaller $t$ reaches $5.$ Thus $T=12,$ and the sum of its digits is $1+2=3.$

Thus, the correct answer is B.

2017 AMC 12A Problem 12

Solution:

Problem 12 in Other Years