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CAT 1995 — QA Question 11

Number TheoryEasy

For the product n(n + 1)(2n + 1), n ∈ N, which one of the following is not necessarily true?

Answer & solution

Correct answer: Never divisible by 237

  • A

    It is even

  • B

    Divisible by 3

  • C

    Divisible by the sum of the square of first n natural numbers

  • Never divisible by 237

Solution

Since n(n + 1) are two consecutive integers, one of them will be even and thus their the product will always be even.
Also, sum of the squares of first ‘n’ natural numbers is given by n(n+1)(2n+1)6.

Hence, our product will always be divisible by this.
Also you will find that the product is always divisible by 3 (you can use any value of n to verify this).
However, we can find that option (d) is not necessarily true. E.g. If n = 118, (2n + 1) = 237 or if n = 236, then (n + 1) = 237 or if n itself is 237, etc.

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CAT 1995 QA Q11: For the product n(n + 1)(2n + 1), n ∈ N, which one of the following is not necessarily true? — Solution | TheCATExam