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.
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
\(n(n + 1)\) is divisible by \(2\) as they are consecutive integers.
If \(n\not\equiv 0\) (mod 3) then there arise two casess……..
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 ……….
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
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