Question

Traffic lights at three different roads change after 24,36 and 54 seconds respectively.If they set to change simultaneously at 6.00 am then after how many minutes and seconds they change again simultaneously?

• A. 3 minutes 36 seconds
• B. 3 minutes 45 seconds
• C. 4 minutes 15 seconds
• D. 4 minutes 26 seconds

Hint:

For problems involving repeated events occurring together, always use the LCM of their individual time intervals.

Answer 1

The Correct Option is A

Concept: When several events repeat at different intervals and we need to find when they occur together again, we calculate the Least Common Multiple (LCM) of the intervals.

Step 1: Finding the prime factorization of the intervals.
\[ 24=2^3 \times 3 \] \[ 36=2^2 \times 3^2 \] \[ 54=2 \times 3^3 \]

Step 2: Calculating the LCM.
Taking the highest powers of all prime factors, \[ LCM=2^3 \times 3^3 \] \[ =8 \times 27 \] \[ =216 \text{ seconds} \]

Step 3: Converting into minutes and seconds.
\[ 216=180+36 \] \[ =3 \text{ minutes }36 \text{ seconds} \] Therefore, the lights will change simultaneously again after \[ 3 \text{ minutes }36 \text{ seconds} \] {3 minutes 36 seconds}