Question

Prove that product of two consecutive positive integers is divisible by $2$.

Answer 1

$n(n+1)=n^2+1$

$=\left(\right(n+1)-1)(n+1)$

$=(n+1{)}^2-(n+1)$

So if $(n+1)$ is an even no $(n+1{)}^2$ is even and diff of even is always even is $(n+1)$.

If odd then $(n+1{)}^2$ is odd and diff of odd is always even

So $n(n+1)$ is always even and divisible by $2$