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Is each other the correct spelling or can it also be used as eachother?
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"eachother" doesn't exist, except for the fact that in spoken English the two words may sound like only one word.
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The American Heritage dictionary only lists each other (pronoun)
Longman Dictionary of Contemporary English also only lists each other (pronoun)
Collins Cobuild Advanced Learner's English Dictionary also only lists each other.

I think that settles it. Until someone can find me a correct example or dictionary entry where it is used as one word.

Hope this helps?
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Comments  
Apparently less than 0.0000 % of all words in all books scaned by google are eachother and between 0.0080 - 0.0140% are each other..

Ngram link:
https://books.google.com/ngrams/graph?content=eachother%2C+each+other&year_start=1800&year_end=20...
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