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If $n$ is any odd number greater than $1$, then $n(n^2 – 1)$ is

  1. divisible by $96$ always
  2. divisible by $48$ always
  3. divisible by $24$ always
  4. None of these
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Let, $n = 2m+1; n>1;m \in \mathbb{Z},$ and $m>0$

Also let, $x = n(n^{2}-1)$

$\Rightarrow x = (2m+1)[ \,(2m+1)^{2}-1]$

$\Rightarrow x = (2m+1)(4m^{2}+1+4m-1)$

$\Rightarrow x = (2m+1)(4m^{2}+4m)$

$\Rightarrow \boxed{x = (2m+1)4m(m+1)}$

Now, we can substitute the various value of $m$

  • $m = 1 \Rightarrow x = 3 \times 4 \times 2 = 24$
  • $m = 2 \Rightarrow x = 5 \times 8 \times 3 = 120$
  • $m = 3 \Rightarrow x = 7 \times 12 \times 4 = 336$
  • $\vdots \quad \vdots \quad \vdots$

$\therefore x = n(n^{2}-1)$  is always divisible by $24.$

Correct Answer $:\text{C}$

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