# Show that the product of two consecutive even numbers is divisible by 8 ?

Mar 27, 2017

See below.

#### Explanation:

An even number $n$ has the structure $n = 2 k$ where $k \in \mathbb{Z}$. Now two consecutive even numbers $n , n + 1$ have the structure

$\left(n , n + 1\right) \to \left(2 k , 2 \left(k + 1\right)\right)$ so the final relationship is

$\left\{n \left(n + 1\right) = p\right\} \to \left\{2 k \left(2 \left(k + 1\right)\right) = p\right\}$

or

$4 k \left(k + 1\right) = p$ for $k \in \mathbb{Z}$

NOTE: $\mathbb{Z}$ denotes the set of integers. This denomination derives from the german word for integer. Zahlen.