Question

For the product $n(n+1)(2n+1)$, $n \in \mathbb{N}$, which one of the following is not necessarily true?

• A. It is even
• B. Divisible by 3
• C. Divisible by the sum of the square of first n natural numbers
• D. Never divisible by 237

Hint:

Check properties for all $n$, not just small values, when the question asks “not necessarily true.”

Answer 1

The Correct Option is D

Expression $n(n+1)(2n+1)$ is always even (two consecutive integers), divisible by 3 (among $n, n+1, 2n+1$ one divisible by 3), and equals $6 \times$ sum of squares of first n naturals. It may be divisible by 237 for some $n$, so (d) is not necessarily true.