About this line '每一个我都属于你'- Does每一个我都 mean 'all of me'?
About this line '每一个我都属于你'- If I read it literally it says Every one of me. This doesn't really make sense. So, I am thinking maybe 每一个我都 means 'all of me'?

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臺灣當紅歌星 A-Lin - 每一個我

http://www.youtube.com/watch?v=Jgvu4HvD51g

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there is nothing wrong about grammar but we need to know more to get what it meant.

I think it means "there are many kinds of me for example, me with a happy mood, me with a bad mood, me with a hard-working time, me with a lazy time. every kind of me/all of me belongs to u----每一个我都属于你"

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Yes, you are right. It means "all of me". 
中文是真的可以這樣說，但是若有更完整的上下文，可以更了解它的意思。

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This might be only a part of the sentence. I guess.

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it means "every part of me"

لا تفوّت فرصة تعلّم لغة جديدة وأنت مرتاح في منزلك. تصفّح مجموعتنا المختارة من مدرّسي اللغات ذوي الخبرة وسجّل في درسك الأول الآن!