Math number theory

Question

Three track team members are running on the school track. It takes Jess 4 minutes to complete a lap, jen 5 minutes, and Jill 6 minutes. If they start at the same point at the same time, how long will it be before they are once again together at the starting point? How many laps will each have completed?

Answer 1

If we let n be the number of minutes that goes by when all three of them are at the starting point again, then

n = 0 mod 4

n = 0 mod 5

n = 0 mod 6

n is a multiple of 4, 5, and 6. We're looking first time it happens, hence the least common multiple of 4, 5, and 6. Rewriting these in terms of primes, we get 2², 5, and 2*3. Since the least common multiple is the product of the primes raised to the maximum of exponents between the numbers we wrote in terms of primes, we get 2²*5*3 = 60. The exponents of 2 were 2, 0, and 1 respectively, so the maximum is 2. We treat the rest this way as well. So the answer is 60 minutes, with Jess running 60/4 = 15 laps, Jen running 60/5 = 12 laps, and Jill running 60/6 = 10 laps.