Question

A) show that $n(2n + 1)(7n + 1)$ is divisible by 6 for all integers $n$.

b.find all integers $n$ such that $n(2n + 1)(7n + 1)$ is divisible by 12.

Answer 1

Hi,

A)

`(n(2n+1)(7n+1))/(6) \\ = (14n^3+9n^2+n)/(6) \\ = (12n^3+6n^2)/(6) + (2n^3+3n^2+n)/(6) \\ =2n^3+n^2+ (n(2n^2+3n+1))/(6) \\ = 2n^3+n^2+ (n(n+1)(2n+1))/(6) \\ = 2n^3+n^2+ 1^2+2^2+3^2+...+n^2\ is\ an\ integer.`

B)
Only if n=4*k or n=4*k+1 , k beeing an integer.