Prove: The product of any two cons. int. has the form....

[Mooch22]

Prove: The product of any two consecutive integers has the form 3k or 3k+2 for some integer k.

I'm really needing help with this proof! Here's all that I have so far:

Proof: Let x and x+1 be consecutive integers.

I don't know where to go after this...?

Thank you for your help.

 

[Denis]

Hint: product of 2 consecutive integers is always even.

 

[galactus]

Let n=3m

Then \(\displaystyle n(n+1)=3m(3m+1)\)

\(\displaystyle 9m^{2}+3m\)

\(\displaystyle 3(\underbrace{3m^{2}+m}_{\text{integer k}})\)

So, when n=3m for some integer m, n(n+1) is 3k for some integer k.