Question

the difference between an integer and its cube is divisible by ​

Answer 1

Diff=x3−x=x(x2−1)=x(x+1)(x−1)via (a2−b2)=(a+b)(a−b)

Diff x3x xx21 xx1x1via a2b2abab

=(x−1)⋅x⋅(x+1)x1xx1

So that always turns out to be product of three consecutive integers...which will always have one even number as 2*m and one number divisible by 3 as 3*n. Hence the product will always have minimal factors as 2 and 3...So, it is divisible by 6. That's it!!!