Define the numbers

Let the three consecutive even natural numbers be $2n-2$, $2n$, and $2n+2$, where $n$ is a natural number starting from 2.

$2n-2, 2n, 2n+2$

Calculate the product

The product is $P = (2n-2)(2n)(2n+2) = 2^3(n-1)(n)(n+1) = 8(n^3 - n)$.

$P = 8(n-1)n(n+1)$

Analyze divisibility

The expression $(n-1)n(n+1)$ is the product of three consecutive integers, which is always divisible by $3! = 6$. Therefore, $P$ is divisible by $8 \times 6 = 48$.

$P = 8 \times 6k = 48k$