Two people are jogging around a circular track in the same direction. One person can go completely around the track in 14 min. The second person takes 12 min. If they both start running in the same place at the same time, how long will it take them to be together if they continue to run at this pace?

Expert Answers

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At first glance, this seems like a "distance = rate x time" problem; however, it's actually a problem involving only the Least Common Multiple (LCM). We are not concerned with how far each travels; only with when their rates will allow their position to coincide again. To find the LCM of 14 and 12, we can complete a prime factorization of each.

Here, if we want to find the LCM of 14 and 12, we can produce a prime factorization of each number:

14: 2 x 7 (prime factorization)

12: 2 x 2 x 3 (prime factorization)

The product of the sets of primes with the highest exponent value of both of the two integers produces the following:

7 x 3 x 2^2 = 84, which is the LCM. We can interpret this to mean that it will take 84 minutes for them to reach the same spot.

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