2017 AMC 10A Problem 16
Problem 16 of 25IntermediateNumber Theory
There are horses, named Horse Horse and so on through 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
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Solution
Horse is back at the starting point after minutes exactly when is divisible by . Thus we need the least positive that is divisible by at least five of the integers .
Checking upward, no number below has five divisors from this list: for example, has , has , has , and has .
The number is divisible by and , so the least possible time is . The sum of its digits is .
Thus, B is the correct answer.