A movie theater runs its films continuously. One movie runs for 80 minutes and a second runs for 110
 minutes. The theater has a 5 -minute intermission after each​ movie, at which point the movie is shown again. If both movies start at​ noon, when will the two movies start again at the same​ time?

To find out when the two movies will start again at the same time, we need to calculate the least common multiple (LCM) of the total running time for each movie, including the intermission.

- The first movie runs for 80 minutes, plus a 5-minute intermission, so its total cycle is \( 80 + 5 = 85 \) minutes.
- The second movie runs for 110 minutes, plus a 5-minute intermission, so its total cycle is \( 110 + 5 = 115 \) minutes.

Now, we need to find the LCM of 85 and 115.

### Step 1: Prime factorization
1. \( 85 = 5 \times 17 \)
2. \( 115 = 5 \times 23 \)

### Step 2: LCM
The LCM is the product of the highest powers of all the prime factors involved:
\[
\text{LCM}(85, 115) = 5 \times 17 \times 23 = 1955 \, \text{minutes}.
\]

### Step 3: Convert to hours and minutes
To convert 1955 minutes into hours and minutes:
\[
1955 \, \text{minutes} = 1955 \div 60 = 32 \, \text{hours} \, 35 \, \text{minutes}.
\]

So, 32 hours and 35 minutes after noon is when the two movies will start again at the same time. This would be:
\[
\text{Next alignment:} \, \text{8:35 PM the next day}.
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you find the least common multiple of two numbers?
2. Can you explain how prime factorization helps in calculating the LCM?
3. Why do we need to add the intermission time to each movie's duration?
4. What is the general formula for calculating time intervals in such periodic situations?
5. How could this concept apply to scheduling in other industries?

**Tip:** Prime factorization is a crucial tool in finding the LCM and GCD (Greatest Common Divisor) in various mathematical problems.

A movie theater runs its films continuously. One movie runs for 80 minutes and a second runs for 110 minutes. The theater has a 5-minute intermission after each movie, at which point the movie is shown again. If both movies start at noon, when will the two movies start again at the same time?

This problem explains how to find when two movies with different durations and intermissions start at the same time again. By calculating the least common multiple (LCM) of 85 minutes and 115 minutes using prime factorization, you'll find that the two movies align again after 32 hours and 35 minutes, which is 8:35 PM the next day. Suitable for Grades 8-10.

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