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Math Help - proof

  1. #1
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    proof

    by definition, a~b means there exists a k in Z such that a-b=3k

    now

    if a~a' and b~b' show ab~a'b'

    Note: S=Z=(0,+or-1,+or-2,.......) and
    _ _ __
    a*b = ab
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by natester View Post
    by definition, a~b means there exists a k in Z such that a-b=3k

    now

    if a~a' and b~b' show ab~a'b'

    Note: S=Z=(0,+or-1,+or-2,.......) and
    _ _ __
    a*b = ab
    starting with your assumptions

    a-a'=3n where n is an integer and

    b-b'=3m where m is an integer.

    so a=a'+3n and b=b'+3m

    so

    ab=(a'+3n)(b'+3m)=a'b'+3a'm+3b'n+9mn=a'b'+3(a'm+b'  n+3mn)

    the term a'm+b'n+3mn is an integer by the closer property of the integers.

    so lets call a'm+b'n+3mn=q then

    ab=a'b'+3q so ab~a'b'
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