2017 AMC 12A Problems/Problem 12
Problem
There are horses, named Horse 1, Horse 2, , Horse 10. They get their names from ow many minutes it takes them to run one lap around a circular race track: Horse runs one lap in exactly 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 , in minutes, at which all horses will gain simultaneously be at the starting point is . Let be the least time, in minutes, such that at least of the horses are again at the starting point. What is the sum of the digits of ?
Solution
We know that Horse will be at the starting point after minutes if . Thus, we are looking for the smallest such that at least of the numbers divide . Thus, has at least positive integer divisors.
We quickly see that is the smallest number with at least positive integer divisors, and that are each numbers of horses. Thus, our answer is .
See Also
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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All AMC 12 Problems and Solutions |
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