# 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 . - Mathematics

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 = "𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑖𝑝𝑎𝑛𝑡𝑠" /"𝐻𝐶𝐹 (60,84,108)"
Prime factorization of 60, 84 and 108 is:
60 = 22 × 3 × 5
84 = 22 × 3 × 7
108 = 22 × 33

HCF = product of smallest power of each common prime factor in the numbers = 22 × 3 = 12
Total number of participants = 60 + 84 + 108 = 252
Therefore, minimum number of rooms required = 252/12 = 21
Thus, minimum number of rooms required is 21.

Concept: Euclid’s Division Lemma
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