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AMC 8 Math Olympiad

Integer Problem |ISI-B.stat | Objective Problem 156

Try this beautiful problem from TOMATO useful for ISI B.Stat Entrance from Integer based on divisibility. You may use sequential hints.

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


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……..
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 ……….


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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