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Ella toma tres medicamentos [tex]$M$[/tex] cada 6 horas y un medicamento [tex]$P$[/tex] cada 4 horas. Determine a qué hora volverá a tomar ambos medicamentos simultáneamente después de las 8 a.m. Para resolver esto, encuentre el mínimo común múltiplo entre 6 y 4.

A.
\begin{tabular}{lll|l}
3 & 4 & 6 & 2 \\
3 & 2 & 3 & 2 \\
3 & 1 & 3 & 3 \\
1 & 1 & 1 &
\end{tabular}

MCM [tex]$= 2 \times 3 = 6$[/tex] dentro de 6 horas

B.
\begin{tabular}{lll|l}
3 & 4 & 6 & 2 \\
3 & 2 & 3 & 2 \\
3 & 1 & 3 & 3 \\
1 & 1 & 1 &
\end{tabular}

MCM [tex]$= 2^2 \times 3 = 12$[/tex] dentro de 12 horas

C.
\begin{tabular}{lll|l}
3 & 4 & 6 & 2 \\
3 & 2 & 3 & 2 \\
3 & 1 & 3 & 3 \\
1 & 1 & 1 &
\end{tabular}

MCM [tex]$= 2 \times 2 \times 3 = 12$[/tex] dentro de 12 horas

D.
\begin{tabular}{lll|l}
3 & 4 & 6 & 3 \\
1 & 4 & 6 & 4 \\
1 & 1 & 6 & 6 \\
1 & 1 & 1 &
\end{tabular}

MCM [tex]$= 2 + 2 + 3 = 7$[/tex] dentro de 7 horas

Sagot :

GinnyAnswer
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.
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