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 HCF between two green lights:

Options

A. 18
B. 20
C. 16
D. 22

Answer

To find the Highest Common Factor (HCF) of the two intervals of the traffic lights, which are 90 seconds and 144 seconds, we can use the prime factorization method. Here’s how we can do it step by step:

### Step 1: Prime Factorization of 90
- Start by dividing 90 by the smallest prime number, which is 2.
- 90 ÷ 2 = 45
- Next, divide 45 by the smallest prime number, which is 3.
- 45 ÷ 3 = 15
- Divide 15 by 3 again.
- 15 ÷ 3 = 5
- Finally, 5 is a prime number itself.

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

### Step 2: Prime Factorization of 144
- Start by dividing 144 by the smallest prime number, which is 2.
- 144 ÷ 2 = 72
- Divide 72 by 2 again.
- 72 ÷ 2 = 36
- Divide 36 by 2 again.
- 36 ÷ 2 = 18
- Divide 18 by 2 again.
- 18 ÷ 2 = 9
- Now, divide 9 by the smallest prime number, which is 3.
- 9 ÷ 3 = 3
- Finally, divide 3 by 3.
- 3 ÷ 3 = 1

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

### Step 3: Identify Common Factors
Now, we will identify the common prime factors from the factorizations of 90 and 144:
- For 2: The minimum power is \(2^1\) (from 90).
- For 3: The minimum power is \(3^2\) (common in both).

### Step 4: Calculate HCF
Now, we can calculate the HCF by multiplying the common prime factors with their minimum powers:
\[ HCF = 2^1 \times 3^2 = 2 \times 9 = 18 \]

### Final Answer
The HCF of 90 and 144 is **18**.

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