When we pass from crossing on a road we all see traffic lights blinking there. A traffic controller set the timmings of traffic lights in such a way that all lights are not green at the same time or specially not in the rush hour, because it can create chaos or problems. So, he take the timings of nearby places in same area and calculate LCM of all traffic stops and he easily manage the traffic by increase the duration or set at different times.
There are two traffic lights on a particular highway which shows green light on time of 90 seconds and 144 seconds respectively.
Calculate their LCM.

Options

A. 720
B. 750
C. 725
D. 700

Answer

To calculate the Least Common Multiple (LCM) of the two traffic light timings, 90 seconds and 144 seconds, we can follow these steps:

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

**For 90:**
- Divide 90 by 2:
\( 90 \div 2 = 45 \)
So, \( 90 = 2 \times 45 \)
- Divide 45 by 3:
\( 45 \div 3 = 15 \)
So, \( 45 = 3 \times 15 \)
- Divide 15 by 3:
\( 15 \div 3 = 5 \)
So, \( 15 = 3 \times 5 \)
- Finally, 5 is a prime number.

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

**For 144:**
- Divide 144 by 2:
\( 144 \div 2 = 72 \)
So, \( 144 = 2 \times 72 \)
- Divide 72 by 2:
\( 72 \div 2 = 36 \)
So, \( 72 = 2 \times 36 \)
- Divide 36 by 2:
\( 36 \div 2 = 18 \)
So, \( 36 = 2 \times 18 \)
- Divide 18 by 2:
\( 18 \div 2 = 9 \)
So, \( 18 = 2 \times 9 \)
- Divide 9 by 3:
\( 9 \div 3 = 3 \)
So, \( 9 = 3^2 \)

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

### Step 2: Identify the Highest Powers
Next, we will identify the highest power of each prime factor from both factorizations:
- For \(2\): The highest power is \(2^4\) (from 144).
- For \(3\): The highest power is \(3^2\) (common in both).
- For \(5\): The highest power is \(5^1\) (from 90).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:
\[ \text{LCM} = 2^4 \times 3^2 \times 5^1 \]

Calculating this step-by-step:
1. \(2^4 = 16\)
2. \(3^2 = 9\)
3. \(5^1 = 5\)

Now, multiply these results:
\[ \text{LCM} = 16 \times 9 \times 5 \]

Calculating \(16 \times 9\):
\[ 16 \times 9 = 144 \]

Now multiply by 5:
\[ 144 \times 5 = 720 \]

Thus, the LCM of 90 seconds and 144 seconds is:
\[ \text{LCM} = 720 \]

### Final Answer
The LCM of the two traffic light timings is **720 seconds**.

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