Question

find any positive integer prove that ncube-n is divisible by 6

Answer 1

n3−n=n(n2−1)=n(n+1)(n−1)=(n−1)n(n+1)∴n3−n=n(n2−1)=n(n+1)(n−1)=(n−1)n(n+1)

The above number: (n−1)n(n+1)(n−1)n(n+1) is the product of three consecutive positive integers (n≥2)(n≥2) which is divisible by 3!=63!=6

Hence, the number: n3−nn3−n is divisible by 66 for all positive integers n

Answer 2

if simplify it then u will get a multiplication of three consecutive no.s and for +ve integers multiplication of three consecutive no.s is always a divisible of 6