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In a seminar, the number of participants in Hindi, English, and Mathematics 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.

Last updated date: 17th Jul 2024
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Hint: We have to find a minimum number of participants. For that, you need to find the total number of participants. Also, find the HCF of $60,84$ and $108$. Use the formula\[Number\text{ }of\text{ }rooms\text{ }required=\dfrac{total\text{ }number\text{ }of\text{ }participants}{12}\].

Complete step-by-step answer:
The greatest number which divides each of the two or more numbers is called HCF or Highest Common Factor. It is also called the Greatest Common Measure(GCM) and Greatest Common Divisor(GCD). HCF and LCM are two different methods, whereas LCM or Least Common Multiple is used to find the smallest common multiple of any two or more numbers.
Follow the below-given steps to find the HCF of numbers using the prime factorization method.
Step 1: Write each number as a product of its prime factors. This method is called here prime factorization.
Step 2: Now list the common factors of both the numbers.
Step 3: The product of all common prime factors is the HCF ( use the lower power of each common factor).
The largest number that divides two or more numbers is the highest common factor (HCF) for those numbers. For example, consider the numbers $36({{2}^{2}}\times {{3}^{2}}),42(2\times 3\times 7)$. $3$ is the largest number that divides each of these numbers, and hence, is the HCF for these numbers.
HCF is also known as Greatest Common Divisor (GCD)
To find the HCF of two or more numbers, express each number as a product of prime numbers. The product of the least powers of common prime terms gives us the HCF.
In the question, the number of rooms will be minimum if each room accommodates the maximum number of participants, since in each room, the same number of participants are to be seated and all of them must be of the same subject.

Therefore, the number of participants in each room must be the HCF of $60,84$ and $108$ .

The prime factorizations of $60,84$ and $108$ are as under.

$60={{2}^{2}}\times 3\times 5$
$84={{2}^{2}}\times 3\times 7$
$108={{2}^{2}}\times {{3}^{2}}$

So HCF for $60,84$ and$108$ is ${{2}^{2}}\times 3=12$.

\[Number\text{ }of\text{ }rooms\text{ }required=\dfrac{total\text{ }number\text{ }of\text{ }participants}{12}\]
\[Number\text{ }of\text{ }rooms\text{ }required=\dfrac{60+84+108}{12}\]

\[Number\text{ }of\text{ }rooms\text{ }required=\dfrac{252}{12}=21\]
So the minimum number of rooms is $21$.

Note: Carefully read the question. You should know the concepts related to HCF. Also, you must know that HCF can be solved by two methods: prime factorization and division method. Don’t make confusion while writing the HCF. Don’t miss any of the terms. In division method divide the largest number by the smallest number of the given numbers until the remainder is zero. The last divisor will be the HCF of given numbers.