SOLUTION: show that 3 divides exactly any one of n,n+1 or n+3?

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Question 977112: show that 3 divides exactly any one of n,n+1 or n+3?
Found 2 solutions by Alan3354, solver91311:
Answer by Alan3354(69443) (Show Source):
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Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

I think you meant the claim to be that 3 divides one of n, n + 1, or n + 2 since if 3 divides n it must, perforce divide n + 3.

Claim: For all integers n, 3 divides exactly one of n, n + 1, or n + 2

We prove by exhaustive cases using the Generalized Divisibility Theorem

Generalized Divisibility Theorem

such that

and

From which it follows that all integers can be represented by one of the forms:

for for some integer

Cases:

The three cases are exhaustive and each case has the same conclusion, hence the claim is proven.

John

My calculator said it, I believe it, that settles it