Proving 3 Divides at Least One Integer

[dgentile]

Homework Statement

Prove that 3 divides one of the integers n, n + 2, or n + 4, for any integer n.

Homework Equations

The Attempt at a Solution

 

[Dick]

Try, ok? Any integer n has a remainder of either 0, 1 or 2 when divided by 3. Come on.

 

[dgentile]

Okay...so what's the formula for proving that true is my question?

Thanks

 

[Dick]

Read the forum rules. You are supposed to put something in the "attempt at a solution section". I just gave you a hint. There really is no 'formula'.

 

[sutupidmath]

Let n be any such integer. THen by the division algorithm there exist integers q and r such that

$$n=3q+r,$$

$$n+2=3q_1+r$$

$$n+4=3q_2+r$$ $$0\leq r<3$$

i.e r=0, 1 or 2

Now, say if r=0, what happens? if r=1, what happens? if r=2 what happens?

P.S. This is what Dick said, i just wanted to make a little bit easier on you.
YOu MUST show your work next time.