To determine when four gongs striking at intervals of 30, 60, 90, and 150 minutes will strike together again, we need to find the Least Common Multiple (LCM) of these four intervals.
### Steps to solve:
1. Prime Factorization:
First, we perform the prime factorization of each interval:
- 30: [tex]\(30 = 2 \times 3 \times 5\)[/tex]
- 60: [tex]\(60 = 2^2 \times 3 \times 5\)[/tex]
- 90: [tex]\(90 = 2 \times 3^2 \times 5\)[/tex]
- 150: [tex]\(150 = 2 \times 3 \times 5^2\)[/tex]
2. LCM Calculation:
The LCM is found by taking the highest power of each prime factor that appears in these factorizations:
- For 2: The highest power is [tex]\(2^2\)[/tex] (from 60).
- For 3: The highest power is [tex]\(3^2\)[/tex] (from 90).
- For 5: The highest power is [tex]\(5^2\)[/tex] (from 150).
Therefore,
[tex]\[
\text{LCM} = 2^2 \times 3^2 \times 5^2
\][/tex]
3. Multiplying the Factors:
- [tex]\(2^2 = 4\)[/tex]
- [tex]\(3^2 = 9\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
Now multiply these together:
[tex]\[
\text{LCM} = 4 \times 9 \times 25
\][/tex]
[tex]\[
= 4 \times 225
\][/tex]
[tex]\[
= 900
\][/tex]
So, the LCM of 30, 60, 90, and 150 minutes is 900 minutes.
4. Convert Minutes to Hours and Minutes:
900 minutes is:
- [tex]\( \frac{900}{60} = 15 \)[/tex] hours
5. Calculate the Time:
Since the gongs start striking together at 12 noon,
[tex]\[
12 \text{ noon} + 15 \text{ hours} = 3 \text{ AM (next day)}
\][/tex]
Therefore, the four gongs will strike together again at 3 AM the next day.
