Question

If n is an even natural number, then what is the least possible factor of the expression $n(n+1)(n+2)$?

• A. 6
• B. 12
• C. 24
• D. 48

Answer 1

The correct option is C 24

$n$ is even $\Rightarrow (n+2)$ is even
Which means, n and n+2 are consecutive even numbers.
In a set of consecutive even numbers, one of them will be divisible by 2 and the other will be divisible by 4.
Thus, $n(n+2)$ is divisible by $2\times 4=8$

Also, $n,(n+1)$ and $(n+2)$ are 3 consecutive numbers, which means that one of them is definitely divisible by 3.

Thus, $n(n+1)(n+2)$ is divisible by $8\times 3=24$

Hence, 24 is the least possible factor for $n(n+1)(n+2)$ if n is an even natural number.