What is the GCD of all numbers of the form $i \times(i+1) \times(i+2)$ with $i \geq 1$ (we are looking for the greatest common divisor of all these numbers).
For $i=1$ we have $i \times(i+1) \times(i+2)=6$. Therefore, the GCD must be less than 6.

But among three consecutive numbers, there is always a multiple of 2 and a multiple of 3, so $i \times(i+1) \times(i+2)$ is always a multiple of 6.

The GCD of the numbers $i \times(i+1) \times(i+2)$ for $i \geq 1$ is exactly 6.