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a

$400|N$

b

$600|N$

c

$8100|N$

d

$N$ is divisible by four distinct prime numbers

detailed solution

Correct option is D

First and second prizes in Maths and Physics can be awarded in

${}^{30}{P}_{2}{.}^{30}{P}_{2}={\left({}^{30}{P}_{2}\right)}^{2}$

First prize is Chemistry, Biology can be awarded in $=30.30={\left(30\right)}^{2}$

$\therefore $

As $400={2}^{4}{.5}^{2},\text{\hspace{0.17em}}600={2}^{3}{\mathrm{.3.5}}^{2},\text{\hspace{0.17em}}8100={2}^{2}{.3}^{4}{.5}^{2},$

We get N is divisible by each of $400,\text{\hspace{0.17em}}600,\text{\hspace{0.17em}}8100$

and also N is divisible by 4 primes 2, 3, 5, 29

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