Chapter 9.5, Problem 46E

Solution

To prove : An alternative proof of the assertion based on the fact that $n\left(n+1\right)\left(n+2\right)$ is product of two consecutive integers.

Given information :

The statement $n\left(n+1\right)\left(n+2\right)$ .

Formula used :

Use the fact that one of the consecutive integers is always even and other is odd and the third one will be divisible by 3.

Explanation :

Consider the statement $n\left(n+1\right)\left(n+2\right)$ ,

These are three consecutive integers, so at least one of these must be even, one must be multiple of three.

This implies that product is divisible by 6.

Thus, 6 is a factor of $n\left(n+1\right)\left(n+2\right)$ .