Try this beautiful problem Based on Integer, useful for ISI B.Stat Entrance.

## Integer | ISI B.Stat Entrance | Problem 156

Let n be any integer. Then \(n(n + 1)(2n + 1)\)

- (a) is a perfect square
- (b) is an odd number
- (c) is an integral multiple of 6
- (d) does not necessarily have any foregoing properties.

**Key Concepts**

Integer

Perfect square numbers

Odd number

## Check the Answer

But try the problem first…

Answer: (c) is an integral multiple of 6

TOMATO, Problem 156

Challenges and Thrills in Pre College Mathematics

## Try with Hints

First hint

\(n(n + 1)\) is divisible by \(2\) as they are consecutive integers.

If \(n\not\equiv 0\) (mod 3) then there arise two casess……..

Case 1,,

Let \(n \equiv 1\) (mod 3)

Then \(2n + 1\) is divisible by 3.

Let \(n \equiv2\) (mod 3)

Then\( n + 1\) is divisible by \(3\)

Can you now finish the problem ……….

Final Step

Now, if \(n\) is divisible by \(3\), then we can say that \(n(n + 1)(2n + 1)\) is always

divisible by \(2*3 = 6\)

Therefore option (c) is the correct

## Other useful links

- https://www.cheenta.com/problem-based-on-triangle-prmo-2018-problem-13/
- https://www.youtube.com/watch?v=pLAMlNUOdTs

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