A quick riddle in number theory

[Yoni]

This question appeared in a national mathematics olympiad a few years ago.
It was the easiest question on the test.

I remember that when I participated, I couldn't solve this one. But now I was looking through old papers, I found this question again, thought about it for 3 minutes - and solved it.

Here is the riddle in its original phrasing (translated to English).

"For which natural numbers n will the numbers n+1, n+11, n+111 all be prime?"

(Hint: The solution is quick and elegant.)

For anyone who's interested but can't solve it right away, please go ahead and post your comments/thoughts anyway.

[Mr. Neo]

n=2
2+1=3, Prime.
2+11=13, Prime.
2+111=113, Prime.

That's the only one that I could find, perhaps I am missing a few.

[Meh]

could N= any even number?

[UserLoser.]

n=0

[Hitmen]

n=0

1 is not a prime number.

[Mr. Neo]

could N= any even number?

No, take 6 for an example.  6+1 is 7, which is a prime number.  11+6 is 17, which is also a prime number, but 111+6=117 which is not a prime number.

[K]

n=0

111 / 3 = 37.

could N= any even number?

11 + 4 = 15.
15 / 3 = 5.

[Adron]

Quick answer, check the first line for big clue...

n + 1, n + 2 + 3*3, n + 3*37

One of those will be divisible by three, so the only possibility is when that one is equal to three (since three itself is prime).

[iago]

Incidentally, Natural numbers are 1+, Whole numbers are 0+, and Integers are positive or negative with no decimal (I forget how it's defined).

[TheMinistered]

n=infinity

[Yoni]

n=infinity

Hmmmmmmmmmmmmm, no.

(Adron's post is the answer, read it for a spoiler)

[Slaughter]

Yoni, I recently returned to the forums, sorry about this belated post - I understand the answer, but not how you came to the conclusion.