To determine when Ella will take both medications simultaneously, we need to calculate the least common multiple (LCM) of their respective intervals. Ella takes medication [tex]\( M \)[/tex] every 6 hours and medication [tex]\( P \)[/tex] every 4 hours. Let's outline the steps to find the LCM:
1. Identify the Intervals:
- Medication [tex]\( M \)[/tex]: every 6 hours
- Medication [tex]\( P \)[/tex]: every 4 hours
2. Prime Factorization:
- Factorize each interval into its prime factors.
- [tex]\( 6 = 2 \times 3 \)[/tex]
- [tex]\( 4 = 2^2 \)[/tex]
3. Determine the Highest Powers of Each Prime:
- For [tex]\( 2 \)[/tex]: the highest power is [tex]\( 2^2 \)[/tex] (from 4).
- For [tex]\( 3 \)[/tex]: the highest power is [tex]\( 3 \)[/tex] (from 6).
4. Calculate the LCM:
- Multiply the highest powers of all prime factors.
- [tex]\( LCM = 2^2 \times 3 = 4 \times 3 = 12 \)[/tex]
Thus, Ella will take both medications simultaneously every 12 hours.
Given that she starts taking the medications at 8 a.m., the next time she will take both medications together will be 8 a.m. + 12 hours = 8 p.m.
In summary:
- Ella will take both medications [tex]\( M \)[/tex] and [tex]\( P \)[/tex] simultaneously every 12 hours.
- If she starts at 8 a.m., the next simultaneous dose will be at 8 p.m.
