The initial question gives the following information: Marcy takes two kinds of medicine; one every 4 hours, and the other every 6 hours. We are told that Marcy took both medications at 10 a.m., and we are asked to find when Marcy will take both medications.

This problem can be solved in a number of ways. You can always use a brute force approach: list the times for each medication and note when they are the same.

M1: 10a,2p,6p,10p,2a,6a,10a,2p,6p,10p,2a, etc.

M2: 10a,4p,10p,4a,10a,4p.10p,4a,10a, etc

You should notice that the first time the medications are taken together is 10pm; the next time is the next morning at 10am; then again the next day at 10pm. **So they are taken together every 12 hours.**

A more efficient approach is to recognize that you need the least common multiple of 4 and 6. To find the LCM, we factor each number, then the LCM is the product of every factor in either number raised to the highest power in either factorization. Ex:

`4: 2*2=2^2`

`6: 2*3`

The factors of the LCM are 2 and 3. 2 will be squared, as that is the highest power, and 3 is to the first power.

So the LCM(4,6)=`2^2*3=12`

It is not a coincidence that we get 12 and the times the medications are taken together are 12 hours apart.

Other problems of this type include:

1) Running/biking around a track at different speeds (easiest if the speeds are in laps per hour).

2) Pizza ovens are dedicated to cooking pizzas with different cooking times (e.g., 15 min and 20 min).

3) Replacing fluids in a car (e.g., change oil every 5000 miles and flush/refill radiator every 40000 miles).

All of these problems have something in common—two activities occur at two different rates, and the situation involves cycles (e.g. time on a clock, going around a cyclical path, etc.).

**Further Reading**