Question

If n is an integer greater than 6, which of the following must be divisible by 3 ?

• A. $n(n+1)(n-4)$
• B. $n(n+2)(n-1)$
• C. $n(n+3)(n-2)$
• D. $n(n+4)(n-2)$
• E. $n(n+5)(n-6)$

Answer 1

The correct option is A $n(n+1)(n-4)$

$n(n+1)(n-4)=n(n+1)(n-4+2-2)=n(n+1)(n+2-6)=n(n+1)(n+2)-6n(n+1)$

One of every three consecutive integers is divisible by $3$ therefore product of three consecutive integers is divisible by $3$

$\therefore n(n+1)(n+2)$ is divisible by $3$

$6$ is divisible by $3$

$\therefore 6n(n+1)$ is divisible by $3$

$\Rightarrow n(n+1)(n-4)$ is divisible by $3$

So option $A$ is correct.