prove that the product of every three consecutive integers is always divisible by 3 as well as 2.

Lets take three consecutive integers as n , n+1 and n+2. Now, there are 2 possibilities - 

1. Two numbers are odd and one is even. 

2. Or two numbers are even and one is odd.

So the even number (irrespective of the fact that there would be 1 or 2 even numbers) is always divisible by two.

 

And one of the odd numbers is divisible by three (remember you are taking three consecutive numbers and every third integer is a number series is divisible by 3). Therefore the product is divisible by 2 as well as 3.