Joe P. answered • 11/24/19

BS in Math with 10+ Years of Research in and Tutoring Discrete Math

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 2^{2}, 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 2^{2}*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.

Stanton D.

Succinct and elegant.01/21/20