Which is correct?
I`m going away at the end of january or in the end of january?

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Answer

"At the end of January" is correct.

The preposition is refering to " the end". It is not refering to "January". In addition, "january" needs to be capitalized.

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my opinion:
at-----------a point directly.  appointed time
in------------not a point.  like the meaning of "during".

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I'm going away at the end of January. --- This is the correct expression. At least in the United States. 

Here are some other examples that might help.

I'm going away in January.
I'm going away at the beginning of January.
I'm going away in the middle of January.

I'm hope this helps. And I hope you have a good trip!  :]

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Prepositions of time

AT the end of January. (specific) - at the beginning of the month
IN January (months)
IN 2011 (years)
ON Monday (days of the week)
AT 3pm (times)

ALWAYS.

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i think both are correct