Answer :
To answer these questions, we need to calculate the Least Common Multiple (LCM) of the given numbers. The LCM of a set of numbers is the smallest number that is divisible by each of them.
Bus Departure Schedule
- The buses on routes 1, 2, and 3 run every 25 minutes, 30 minutes, and 40 minutes, respectively.
- To find when they will all depart together again, we calculate the LCM of 25, 30, and 40.
- First, find the prime factorization of each number:
- 25 = [tex]5^2[/tex]
- 30 = [tex]2 \times 3 \times 5[/tex]
- 40 = [tex]2^3 \times 5[/tex]
- The LCM is obtained by taking the highest power of each prime present in the factorizations:
- LCM = [tex]2^3 \times 3^1 \times 5^2 = 600[/tex]
- Therefore, all three buses will depart together every 600 minutes, or 10 hours.
Athletes on a Circular Track
- The lap lengths are 120 m, 150 m, 180 m, and 240 m.
- Find the LCM of these lengths to determine when they will be together again.
- Prime factorizations:
- 120 = [tex]2^3 \times 3 \times 5[/tex]
- 150 = [tex]2 \times 3 \times 5^2[/tex]
- 180 = [tex]2^2 \times 3^2 \times 5[/tex]
- 240 = [tex]2^4 \times 3 \times 5[/tex]
- LCM = [tex]2^4 \times 3^2 \times 5^2 = 3600[/tex]
- The athletes will all be together again after 3600 meters.
School Classes Schedule
- Classes start every 28 days, 35 days, and 42 days.
- We need to find the LCM of 28, 35, and 42.
- Prime factorizations:
- 28 = [tex]2^2 \times 7[/tex]
- 35 = [tex]5 \times 7[/tex]
- 42 = [tex]2 \times 3 \times 7[/tex]
- LCM = [tex]2^2 \times 3^1 \times 5^1 \times 7^1 = 420[/tex]
- Therefore, all classes will start on the same day again after 420 days.
By finding the LCM, we can determine when events that occur at different intervals will coincide again.