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For the product $n(n + 1)(2n + 1)$, $n \hat{I} N$, which one of the following is necessarily false?

  1. It is always even
  2. Divisible by $3$.
  3. Always divisible by the sum of the square of first n natural numbers
  4. Never divisible by $237$.
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1 Answer

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Clearly the expression is basically the formula for sum of square of 1st 'n' natural numbers  but without the "6" in the denominator.

So options 1,2,4 are true.

But what about 237 ?

The above expression will obviously be divisible by 237 when n= 237 as it will be 237*238*475 In fact even for n = 236 and n = 118 it will be divisible by 237

Option C is the Correct Answer.

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