Understanding the Least Common Multiple (LCM) for Exercise Schedules

Ben exercises every 12 days and Isabel exercises every 8 days. Since they both exercised today, we need to determine how many days it will be until they exercise together again. This involves finding the least common multiple (LCM) of their exercise schedules.

Defining the Least Common Multiple (LCM)

The least common multiple (LCM) of two integers is the smallest positive integer that is evenly divisible by both integers. Mathematically, for two integers a and b, denoted as LCM(a, b), it is the smallest positive integer that is divisible by both a and b. For example, LCM(2, 3) 6 and LCM(6, 10) 30.

Calculating the LCM of 12 and 8

To find the LCM of 12 and 8, we start by listing their multiples until we find the lowest common multiple:

12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120...

8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120...

From the list, we can see that the first common multiple is 24. Therefore, the LCM of 12 and 8 is 24.

Implications for Exercise Schedules

Now, let's apply this understanding to Ben and Isabel's exercise schedules:

Ben's Schedule:

Today: 0 days 24 days: 2 times 48 days: 4 times 72 days: 6 times

Isabel's Schedule:

Today: 0 days 24 days: 3 times 48 days: 6 times 72 days: 9 times

Therefore, they will exercise together again in 24 days. Further, they will exercise together every 24 days after that, as the LCM of 12 and 8 dictates a 24-day interval.

Further Applications of LCM in Exercise Planning

The LCM is a useful tool in planning exercise schedules for teams and individuals. By ensuring that exercise schedules overlap, it can help in setting a common schedule for group activities or to avoid conflicting schedules.

Conclusion

In summary, the LCM of 12 and 8 is 24. Ben and Isabel will exercise together again in 24 days, and they will do so every 24 days afterward. Understanding LCM in the context of exercise schedules can greatly help in planning and coordination.

Key Takeaways:

LCM is the smallest positive integer divisible by both integers. LCM(12, 8) 24. Ben and Isabel will exercise together every 24 days.