15. Team Green meets at the movie every 3 days, Team Yellow meets at the movie every 2 days, and Team Red meets at the movie every 4 days. When will all three teams meet at the movie on the same day?
We need to find the first day that is a multiple of 3, 2, and 4.
One way to do this is to start listing out the multiples of each number and look for the smallest one that they all have in common:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60...
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60...
The smallest number they all have in common is 12. So the three teams will next meet at the movie on the 12th day.
To determine when all three teams will meet at the movie on the same day, we need to find the least common multiple (LCM) of the intervals at which each team meets.
The intervals at which the teams meet are 3 days, 2 days, and 4 days.
To find the LCM of these numbers:
1. List the prime factors of each number:
- 3: 3
- 2: 2
- 4: 2, 2
2. Identify the highest power for each prime factor:
- 3 (highest power: 1)
- 2 (highest power: 2)
3. Multiply the prime factors raised to their highest powers: 3^1 * 2^2 = 12.
Therefore, all three teams, Green, Yellow, and Red, will meet at the movie on the same day every 12 days.