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): You can put this solution on YOUR website!
Duplicate, and it doesn't
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eg
4, 5 or 7
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
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