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Question

In a seminar number of participants in Hindi , English and Mathematics are $$60 , 84 $$ and $$108$$ respectively. If the equal number of participants of the same subject are sitting in each room, then find the least number of required rooms.


Solution

Since the least numbers of a room are required.
So, number of participants in each room must be the H.C.F of $$60 , 84$$ and $$108$$.
$$60 = 2^{2} \times 3 \times 5 $$
$$84 = 2^{2} \times 3 \times 7$$
and $$108 = 2^{2} \times 3^{3} $$ 
HCF of $$60 , 84$$ and $$108 = 2^{2} \times 3 = 4 \times 3 = 12$$.
Therefore, in each room $$12$$ participants can be seated.
$$∴$$ Number of rooms required $$=\dfrac{Total\ number\ of\ participants }{H.C.F}$$
$$=\dfrac{60+84+108}{12}=\dfrac{252}{12}=21$$
Hence $$21$$ room are needed.

Maths

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