The traffic lights of three cross roads change after each 1min,75s and 90s respectively IF these lights change simultaneously at `7:50am` then they will change simultaneously again at

Options

A. `7:59 am`
B. `8:02am`
C. `8:05 am`
D. `8:10am`

Answer

To solve the problem of when the three traffic lights will change simultaneously again after initially changing at 7:50 AM, we need to find the least common multiple (LCM) of the time intervals at which each light changes.

### Step-by-step Solution:

1. **Convert all time intervals to seconds:**
- The first traffic light changes every 1 minute, which is \(60\) seconds.
- The second traffic light changes every \(75\) seconds.
- The third traffic light changes every \(90\) seconds.

So, we have the time intervals: \(60\) seconds, \(75\) seconds, and \(90\) seconds.

**Hint:** Always convert time intervals to the same unit for easier calculations.

2. **Find the LCM of the time intervals:**
- To find the LCM, we can use the prime factorization method.
- \(60 = 2^2 \times 3^1 \times 5^1\)
- \(75 = 3^1 \times 5^2\)
- \(90 = 2^1 \times 3^2 \times 5^1\)

- Now, take the highest power of each prime factor:
- For \(2\): \(2^2\) (from \(60\))
- For \(3\): \(3^2\) (from \(90\))
- For \(5\): \(5^2\) (from \(75\))

- Now, calculate the LCM:
\[ \text{LCM} = 2^2 \times 3^2 \times 5^2 = 4 \times 9 \times 25 \]
\[ = 36 \times 25 = 900 \text{ seconds} \]

**Hint:** The LCM is the smallest number that is a multiple of all the given intervals.

3. **Convert LCM back to minutes:**
- \(900\) seconds can be converted to minutes:
\[ 900 \text{ seconds} = \frac{900}{60} = 15 \text{ minutes} \]

**Hint:** Remember to convert seconds back to minutes if needed for the final answer.

4. **Calculate the next simultaneous change time:**
- The traffic lights change simultaneously at \(7:50 AM\).
- Adding \(15\) minutes to \(7:50 AM\):
\[ 7:50 AM + 15 \text{ minutes} = 8:05 AM \]

**Hint:** When adding time, ensure you account for the transition from AM to PM if necessary.

### Final Answer:
The three traffic lights will change simultaneously again at **8:05 AM**.