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:
Factor tree is used for determining the:

Options

A. HCF
B. LCM
C. prime factor
D. None of these

Answer

To solve the question regarding the traffic lights that show green at intervals of 90 seconds and 144 seconds, we need to determine the prime factors of these two numbers using a factor tree.

### Step-by-Step Solution:

1. **Find the Prime Factorization of 90:**
- Start by dividing 90 by the smallest prime number, which is 2.
- \( 90 \div 2 = 45 \)
- Now, factor 45. The smallest prime number that divides 45 is 3.
- \( 45 \div 3 = 15 \)
- Factor 15. Again, the smallest prime number is 3.
- \( 15 \div 3 = 5 \)
- Now, 5 is a prime number and cannot be divided further.
- Thus, the prime factorization of 90 is:
\[ 90 = 2 \times 3^2 \times 5 \]

2. **Find the Prime Factorization of 144:**
- Start by dividing 144 by the smallest prime number, which is 2.
- \( 144 \div 2 = 72 \)
- Continue factoring 72 by 2.
- \( 72 \div 2 = 36 \)
- \( 36 \div 2 = 18 \)
- \( 18 \div 2 = 9 \)
- Now, factor 9. The smallest prime number is 3.
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)
- Thus, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

3. **Identify the Prime Factors:**
- From the factorizations:
- For 90: \( 2, 3, 5 \)
- For 144: \( 2, 3 \)
- The common prime factors are \( 2 \) and \( 3 \).

4. **Determine the Least Common Multiple (LCM):**
- To find the LCM, take the highest power of each prime factor from both numbers.
- For 2: the highest power is \( 2^4 \) (from 144).
- For 3: the highest power is \( 3^2 \) (from both).
- For 5: the highest power is \( 5^1 \) (from 90).
- Therefore, the LCM is:
\[ LCM = 2^4 \times 3^2 \times 5^1 = 16 \times 9 \times 5 \]
- Calculating this gives:
\[ 16 \times 9 = 144 \]
\[ 144 \times 5 = 720 \]

5. **Conclusion:**
- The LCM of 90 and 144 is 720 seconds, which means both traffic lights will be green at the same time every 720 seconds.