We express each interval as a product of prime factors. For 24, we have 24 = 2 × 2 × 2 × 3 = 2³ × 3. For 36, we have 36 = 2 × 2 × 3 × 3 = 2² × 3².

to traffic light change every 24 seconds and 36 seconds respectively change together at 10 o'clock a.m. after how many seconds will I change together again

Answer: 72 seconds

Step-by-step solution

Step 1: Identify the Problem Type

The two traffic lights change at intervals of $24$ seconds and $36$ seconds. They will change together again at a time that is a common multiple of both $24$ and $36$ . Therefore, the required time in seconds is the Least Common Multiple, $\text{LCM}(24, 36)$ .

Step 2: Find the Prime Factorization

We express each interval as a product of prime factors. For $24$ , we have $24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3$ . For $36$ , we have $36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$ .

Step 3: Calculate the LCM

To calculate the LCM, we take the greatest power of each prime factor involved in the numbers. The highest power of $2$ is $2^3 = 8$ , and the highest power of $3$ is $3^2 = 9$ . Multiplying these gives $8 \times 9 = 72$ .

Step 4: State the Final Result

The least common multiple of $24$ and $36$ is $72$ . Thus, the traffic lights will change together again after $72$ seconds (which is $1$ minute and $12$ seconds, at $10:01:12$ a.m.).

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