use the quotient remainder theorem with d=3 to prove that the product of any two consecutive integers has the form 3k or 3k+2 for some integer k

Question

I am trying to figure out how to do this proof using the quotient remainder theorem

Answer 1

Each integer n falls into one of three equivalence classes modulo 3 (We are using quotient remainder theorem implicitly here): n=3j or n=3j+1 or n=3j+2. In the first case n(n+1)=3j(3j+1)=3k (where k=j(3j+1)). In the second case n(n+1)=(3j+1)(3j+2)=9j²+9j+2=3k+2 (where k=3j²+3j). In the third case n(n+1)=(3j+2)(3j+3)=3(3j+2)(j+1)=3k (where k=(3j+2)(j+1)).

 

I hope this helps.