Question

The difference between an $+$ve integer and its cube, is divisible by

• A. $4$
• B. $6$
• C. $9$
• D. None of these

Answer 1

Answer: (B)

$6$

Let $n$ be the positive integer.

Now its cube is $n^3$

We have to check for divisibility of $n^3-n$

$\Rightarrow n^3-n=n(n-1)(n+1)=(n-1)n(n+1)$

This clearly is the product of $3$ consecutive numbers.

The product of $3$ consecutive numbers is divisible by $6$.

As per the divisibility rule for $6,$ the number should be divisible both by $2$ and $3.$

In the product of $3$ consecutive numbers, the number will turn out to be

divisible by both $2$ and $3.$

This can be proved by Mathematical Induction.