Final answer:
To find out when three movies that play every 14th, 15th, and 18th day will show again on the same day, we must calculate the least common multiple (LCM) of 14, 15, and 18. The LCM is 630, meaning they will all be shown again together on day 632 of the festival.
Step-by-step explanation:
The question involves finding a common multiple of three numbers, which is a mathematical problem of number theory and combinatorics. Specifically, we're looking for the least common multiple (LCM) of 14, 15, and 18, which represents the next time the movies will all be played on the same day again at the film festival after day two.
Step-by-step Explanation
1. First, we list the prime factors of each number:
o 14 = 2 × 7
o 15 = 3 × 5
o 18 = 2 × 3 × 3 (32)
2. Next, we identify the unique prime factors involved and their highest powers:
o 2 (from 14 and 18, highest power is 1)
o 3 (from 15 and 18, highest power is 2)
o 5 (from 15, highest power is 1)
o 7 (from 14, highest power is 1)
3. Then, we multiply these together to find the LCM:
o LCM = 21 × 32 × 51 × 71
o LCM = 2 × 9 × 5 × 7
o LCM = 630
4. The movies will therefore all be shown again together on the same day on day 630 of the film festival.
It is important to note that since the movies are shown together on day 2, we are actually looking for the second occurrence of this synchronization, which will be at 630 days after the second day of the festival, making it day 632 in total.