CAT 1995 — QA Question 11
Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.
For the product n(n + 1)(2n + 1), n ∈ N, which one of the following is not necessarily true?
Answer & solution
- 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
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
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.