There are five moons, with lunar cycles of 5, 7, 9, 35, and 180 days. if they all start at new moon, when will they all be full moon at the same time?

To determine when all five moons will be full at the same time, we need to find the least common multiple (LCM) of the lunar cycles of the moons. The moons have lunar cycles of 5, 7, 9, 35, and 180 days.

Finding the LCM of these numbers will tell us the number of days it will take for all five moons to complete full cycles and be full at the same time.

### Steps:
1. **Prime factorization** of each number:
   - 5 = \(5\)
   - 7 = \(7\)
   - 9 = \(3^2\)
   - 35 = \(5 \times 7\)
   - 180 = \(2^2 \times 3^2 \times 5\)

2. **Take the highest powers of all prime factors:**
   - \(2^2\) (from 180)
   - \(3^2\) (from 9 and 180)
   - \(5\) (from 5, 35, and 180)
   - \(7\) (from 7 and 35)

3. **Multiply these highest powers**:
   \[
   \text{LCM} = 2^2 \times 3^2 \times 5 \times 7 = 4 \times 9 \times 5 \times 7
   \]

Let's calculate that.# Calculate the LCM by multiplying the highest powers of the prime factors
lcm = (2**2) * (3**2) * 5 * 7
lcmThe least common multiple (LCM) of the lunar cycles is 1,260 days. This means that all five moons will be full at the same time 1,260 days after they all start at a new moon.

Would you like any further details or have questions?

Here are some related questions:
1. What is the significance of finding the least common multiple?
2. How would the result change if another moon was added with a different cycle?
3. How is the concept of LCM applied in scheduling or calendar systems?
4. Can the LCM be used in problems involving more than cycles, like traffic lights?
5. How could this principle be applied to planetary motion in astronomy?

**Tip:** When dealing with periodic events, the LCM is crucial to finding when all events will align again.

There are five moons, with lunar cycles of 5, 7, 9, 35, and 180 days. If they all start at new moon, when will they all be full moon at the same time?

Discover how to calculate the least common multiple (LCM) of lunar cycles with periods of 5, 7, 9, 35, and 180 days to determine when all five moons will be full at the same time. This problem explores key concepts in number theory and prime factorization, suitable for students in Grades 8-10.

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