Question

The difference between an integer and its cube is divisible by

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

Answer 1

The correct option is B

$6$

Explanation for the correct option :

Let the integer be $x$.

Then its cube will be $x^3$.

So, there difference is $x^3-x$.

Now,

$$\begin{gathered} x^3-x=x\left(x^2-1\right) \\ =x\left(x+1\right)(x-1) \\ =\left(x-1\right)x\left(x+1\right) \end{gathered}$$

Thus $x^3-x$ is a product of three consecutive integers.

We know that the product of three consecutive integers is always divisible by $6$. So, $x^3-x$ is divisible by $6$.

Explanation for the incorrect options :

Option (A) : $4$

Let us consider the integer $2$. Then $2^3-2=6$ is not divisible by $4$. So, option (A) is incorrect.

Option (C) : $9$

Let us consider the integer $3$. Then $3^3-3=24$ is not divisible by $9$. So, option (C) is incorrect.

Hence, option (B) is the correct answer.