possessive and article

0Hi,02br
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00Can you tell me if my reasoning is correct and what the 'the' in the last example is referring to? Most of my knowledge pertaining this is from Mr. M02br
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00a teacher's conference -- a conference for any teachers and 'a' goes with 01i00conference02i00 and 01i00teacher's02i00 acts as a sort of adjective. 02br
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00a teachers' conference -- a conference for all teachers and 'a' goes with 01i00conference02i00 with "teachers'" acting as a sort of adjective.02br
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00the teachers' conference -- a variation of the one above but similar or the same as above in that 'the' goes with conference and "teachers'" acts as an adjective.02br
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00So far so good, I think, but what about this?02br
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01i01u00the02u00 teacher's conference 02i02br
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00What does 'the' modifying as an article here and is it correct that "teacher's" acts as an adjective? 0-