Question

On a road, three consecutive traffic lights change after 36, 42 and 72 seconds. If the lights are first switched on at $9:00$ AM, then at what time will they change simultaneously?
A) $9:08:04$
B) $9:08:24$
C) $9:08:44$
D) $9:08:58$

Answer 1

Hint:
Here, we will first find the LCM of the given three numbers. We will find the factors of the three numbers by using the prime factorization method. We will then multiply all the factors of the three numbers and taking the highest power among the common factors, we will be able to the required L.C.M. of the three numbers. Converting this into minutes, we will be able to find that at what time again, the traffic lights will change simultaneously.

Complete step by step solution:
According to the question, on a road, three consecutive traffic lights change after 36, 42 and 72 seconds
So, first of all, we will find the LCM of these three numbers, i.e. LCM of 36, 42 and 72.
Now, we will write the prime factors of these numbers, thus, we get,
$36 = 2 \times 2 \times 3 \times 3 = {2^2} \times {3^2}$
$42 = 2 \times 3 \times 7$
$72 = 2 \times 2 \times 2 \times 3 \times 3 = {2^3} \times {3^2}$
Now for calculating L.C.M. we take all the factors present in three numbers and the highest power of the common factors respectively. Hence,
L.C.M. of the given 3 numbers \[ = {2^3} \times {3^2} \times 7\]
Applying the exponent on the terms, we get
\[\Rightarrow\] L.C.M. of the given 3 numbers $ = 8 \times 9 \times 7 = 504$
Therefore, the lights will change simultaneously after every 504 seconds
Now, we know that,
$1{\text{ minute}} = 60{\text{ seconds}}$
Hence, by unitary method,
$1{\text{ second}} = \dfrac{1}{{60}}{\text{ minute}}$
Therefore, multiplying both sides by 504, we get,
$504{\text{ seconds}} = \dfrac{{504}}{{60}}{\text{ = 8}}\dfrac{{24}}{{60}}{\text{ minutes}}$
Hence, we get,
$ \Rightarrow 504{\text{ seconds}} = 8{\text{ minutes}}24{\text{ seconds}}$
Thus, if the lights are first switched on at $9:00$ AM, then they will change simultaneously again at:
$9:08:24$

Therefore, option B is the correct answer.

Note:
In this question, we are required to express the given numbers as a product of their prime factors in order to find their LCM. Hence, we should know that prime factors are those factors which are greater than 1 and have only two factors, i.e. factor 1 and the prime number itself.
Now, in order to express the given numbers as a product of their prime factors, we are required to do the prime factorization of the given number. Factorization is a method of writing an original number as the product of its various factors.