When will the three runners be together again?

Question

Three runners start together at the start line of a circular track and run at constant speeds, completing a lap in 4 minutes, 6 minutes and 10 minutes respectively. 1. When are all three next at the start line at the same moment? 2. When are the 4-minute and 6-minute runners next together anywhere on the track (not necessarily at the start)?

Accepted Answer

All three are back at the start together after 60 minutes. The 4-minute and 6-minute runners are side by side every 12 minutes. Part 1. Each runner is at the start line at multiples of their lap time: 4, 8, 12, ...; 6, 12, 18, ...; 10, 20, 30, .... The first time common to all three lists is the least common multiple of 4, 6 and 10. Factor: 4 = 2 squared, 6 = 2 × 3, 10 = 2 × 5, so the LCM is 4 × 3 × 5 = 60 minutes. Part 2. Two runners are together whenever the faster has gained a whole number of laps on the slower. The 4-minute runner completes 1/4 lap per minute, the 6-minute runner 1/6 lap …

When will the three runners be together again?

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