Problem:
There are horses, named Horse , Horse , .., Horse . They get their names from how many minutes it takes them to run one lap around a circular race track: Horse runs one lap in exactly minutes. At time 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 again 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 ?
Answer Choices:
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Solution:
Horse will again be at the starting point after minutes if and only if is a divisor of . Let be the number of integers with that divide . Then , , , and . Thus and the requested sum of digits is .
The problems on this page are the property of the MAA's American Mathematics Competitions