when I write 이천십육 년, do I need to change '육' to '유'?
when I write 이천십육 년, 2016, do I need to change '육' to '유'?

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you are correct , 이천십육년 is right
but people usually don't use '이천십육년' but "2016"

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Here it would be pronounced "이천심늉년"^^ And as Park said, just like in English, they usually write the numerals^^ We write "2016" not "twenty-sixteen" or "two thousand sixteen", right? People want to communicate efficiently - shorter is often sweeter.

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We rerely use '유' for 6. In most cases, we pronounce it '육'.
The only case is that June is the sixth month of year and we say 6월 and pronounce it '유월' - 'YouWol'

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