Question

In a seminar, the number of participants in English, Hindi and Maths are 60, 84 and 108 respectively. Find the minimum number of rooms required if in each room, the same number of participants are to be seated and all of them being in the same subject.

Answer 1

Hint: We will express each number as the product of prime factors and then find the HCF of the numbers, 60, 84 and 108. That would be the number of participants that are to be seated. Then, divide the total number of participants by HCF of the given numbers to find the minimum number of rooms.

Complete step-by-step answer:
The number of participants in English are 60, participants in Hindi are 84 and participants in Maths are 108.
We want to find the number of rooms if the same number of participants are to be seated.
First of all, we will determine the number of participants that should be seated.
The number of participants should be the same.
For finding that number, we will find HCF of 60, 84, 108.
Express each of the numbers as the product of prime factors.

2	60
2	30
3	15
5	5
	1

$60 = 2 \times 2 \times 3 \times 5$

2	84
2	42
3	21
7	7
	1

$84 = 2 \times 2 \times 3 \times 7$

2	108
2	54
3	27
3	9
3	3
	1

$108 = 2 \times 2 \times 3 \times 3 \times 3$
Thus, the HCF of the numbers will be $2 \times 2 \times 3 = 12$\[\]
Hence, each room will have 12 participants.
Divide the total students by the number of participants in one room to find the minimum number of rooms.
$\dfrac{{60 + 84 + 108}}{{12}} = \dfrac{{252}}{{12}} = 21$
Hence, there should be a minimum 21 rooms so that all the participants are of the same number and of the same subject.
Note: HCF stands for highest common factor of the numbers. HCF of numbers divides the numbers completely. Many students make mistakes by finding LCM instead of HCF because of the word minimum written in the question, but this will lead to wrong answers as minimum rooms will be calculated by dividing the total participants by the number of participants in a room.