In a call centre at New Delhi, it is observed that it gets a call at a interval of every 10 minutes from California, at every 12 minutes from Texas, at the interval of 20 minutes from Washington DC and after every 25 minutes it gets the call from London. If in the early morning at 5 : 00 a.m. It has recieved the calls simultanously from all the four destrinations, then at which time it will receive the calls at a time from all the places on the same day?

Options

A. `10 : 00` a.m.
B. `3 : 00` a.m.
C. `5 : 00` p.m.
D. both (a) and (b)

Answer

To solve the problem, we need to find the least common multiple (LCM) of the intervals at which calls are received from the four different locations. The intervals are as follows:

- California: 10 minutes
- Texas: 12 minutes
- Washington DC: 20 minutes
- London: 25 minutes

### Step 1: Identify the intervals
We have the following intervals:
- California: 10 minutes
- Texas: 12 minutes
- Washington DC: 20 minutes
- London: 25 minutes

### Step 2: Find the prime factorization of each interval
- 10 = 2 × 5
- 12 = 2² × 3
- 20 = 2² × 5
- 25 = 5²

### Step 3: Determine the highest power of each prime factor
- For the prime factor 2: The highest power is 2² (from 12 and 20).
- For the prime factor 3: The highest power is 3¹ (from 12).
- For the prime factor 5: The highest power is 5² (from 25).

### Step 4: Calculate the LCM
The LCM is calculated by multiplying the highest powers of all prime factors:
LCM = 2² × 3¹ × 5²
= 4 × 3 × 25
= 12 × 25
= 300 minutes

### Step 5: Convert minutes into hours
To convert 300 minutes into hours, we divide by 60:
300 minutes ÷ 60 = 5 hours

### Step 6: Add the time to the initial call time
The initial simultaneous call time is at 5:00 AM. Adding 5 hours to this time:
5:00 AM + 5 hours = 10:00 AM

### Final Answer
The next time the call center will receive calls simultaneously from all four destinations is at **10:00 AM**.

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