Case Study-2 : In our daily life we all see traffic lights. A traffic controller set the timings of traffic lights is such a way that all light are not green at the same time or specially not in the rush hout. It may create problem in an hour because lights are for few minutes only. So, he take the timmings of nearby places in same area and calculate I cm of all traffic stops and he easily manage the traffic by increasing the duration set at different times. There are two traffic lights on a particular highway which shows green light at the interval of 90 seconds and 144 second respectively.
Read the above paragraph carefully and answer the questions that follows:
Find the LCM between two green lights:

Options

A. 720
B. 730
C. 710
D. 740

Answer

To find the Least Common Multiple (LCM) of the two traffic light intervals, which are 90 seconds and 144 seconds, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of both numbers.

- **For 90:**
- 90 can be divided by 2: \( 90 \div 2 = 45 \)
- 45 can be divided by 3: \( 45 \div 3 = 15 \)
- 15 can be divided by 3: \( 15 \div 3 = 5 \)
- 5 is a prime number.

Thus, the prime factorization of 90 is:
\[ 90 = 2^1 \times 3^2 \times 5^1 \]

- **For 144:**
- 144 can be divided by 2: \( 144 \div 2 = 72 \)
- 72 can be divided by 2: \( 72 \div 2 = 36 \)
- 36 can be divided by 2: \( 36 \div 2 = 18 \)
- 18 can be divided by 2: \( 18 \div 2 = 9 \)
- 9 can be divided by 3: \( 9 \div 3 = 3 \)
- 3 is a prime number.

Thus, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

### Step 2: Identify the Highest Powers of Each Prime
Next, we identify the highest powers of all prime factors from both numbers:

- For the prime number 2, the highest power is \( 2^4 \) (from 144).
- For the prime number 3, the highest power is \( 3^2 \) (common in both).
- For the prime number 5, the highest power is \( 5^1 \) (from 90).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying the highest powers of all prime factors together:

\[ \text{LCM} = 2^4 \times 3^2 \times 5^1 \]

Calculating this step-by-step:

1. Calculate \( 2^4 = 16 \)
2. Calculate \( 3^2 = 9 \)
3. Calculate \( 5^1 = 5 \)

Now multiply these results together:

\[ \text{LCM} = 16 \times 9 \times 5 \]

Calculating \( 16 \times 9 \):

\[ 16 \times 9 = 144 \]

Now multiply by 5:

\[ 144 \times 5 = 720 \]

Thus, the LCM of 90 and 144 is:

\[ \text{LCM} = 720 \]

### Final Answer
The LCM of the two traffic light intervals (90 seconds and 144 seconds) is **720 seconds**.

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