Question

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

Answer 1

Hi,

A)
$\dfrac{n(2n+1)(7n+1)}{6} \\ = \dfrac{14n^3+9n^2+n}{6} \\ = \dfrac{12n^3+6n^2}{6} + \dfrac{2n^3+3n^2+n}{6} \\ =2n^3+n^2+ \dfrac{n(2n^2+3n+1)}{6} \\ = 2n^3+n^2+ \dfrac{n(n+1)(2n+1)}{6} \\ = 2n^3+n^2+ 1^2+2^2+3^2+...+n^2\ is\ an\ integer.$

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