In a seminar, the number of participants in Hindi, English and mathematics are 60, 84 and 108 respectively.

Question:

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.

Solution:

Minimum number of rooms required $=\frac{\text { Total number of participants }}{\operatorname{HCF}(60,84,108)}$

Prime factorization of 60, 84 and 108 is:

$60=2^{2} \times 3 \times 5$

$84=2^{2} \times 3 \times 7$

$108=2^{2} \times 3^{3}$

$\mathrm{HCF}=$ product of smallest power of each common prime factor in the numbers $=2^{2} \times 3=12$

Total number of paricipants = 60 + 84 + 108 = 252

Therefore, minimum number of rooms required $=\frac{252}{12}=21$

Thus, minimum number of rooms required is 21.

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