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Best answer
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The smallest value of n will be 3.

And we can see that n(n2 - 1) = n(n - 1)(n + 1) which is the product of 3 consecutive numbers. For n = 3, This quantity will be 2.3.4 = 24. So the largest number that divides n(n2 - 1) will be 24. 

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As $n(n^{2}-1)$ = (n−1) n (n+1) 

Since n is any odd number , n−1 and n+1 are consecutive even integers.

So, one is divisible by 2 and the other is divisible by 4.

Hence, (n−1) n (n+1) is divisible by 8. 

Since n−1, n , n+1 are 3 consecutive integers, one of them will be divisible by 3.

Hence, (n−1) n (n+1) is divisible by 3. 

Since (n−1) n (n+1) is divisible by both 8 and 3, it is divisible by lcm(8,3)=24.

 

then the largest number that divides  $n(n^{2}-1)$ is 24

Hence Option C is the Correct Answer.

 

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