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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

But try the problem first…

Answer: (c) is an integral multiple of 6

Source

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