$n^3 - n$

Number Theory Level 2

True or False?

$n^3 - n$ is divisible by 3 for every natural $n$ .

can't say false sometimes true true

1 solution

Vilakshan Gupta
Nov 30, 2017

$n^3-n=(n-1)\cdot n \cdot (n+1)$ .

As it is a product of $3$ consecutive numbers, there must be one number which will divide $3$ and hence the given expression is divisible by $3$ .