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Math Help - need help

  1. #1
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    need help

    i need a hint please help
    Show that every positive integer n can be written uniquely in the form n=ab, where a is a square-free and b is a square. Show that b is then the largest square dividing n.
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  2. #2
    Member TheMasterMind's Avatar
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    Quote Originally Posted by mancillaj3 View Post
    i need a hint please help
    Show that every positive integer n can be written uniquely in the form n=ab, where a is a square-free and b is a square. Show that b is then the largest square dividing n.
    By 'square free', do you mean not divisible by any integer of the a^2 with a>1?
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  3. #3
    Moo
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    Quote Originally Posted by TheMasterMind View Post
    By 'square free', do you mean not divisible by any integer of the a^2 with a>1?
    It's the exact definition of square free

    If n is square-free, then a=n and b=1
    If n is a square, then a=1 and b=n

    If n is not square free, nor a square, it means that there exists integers m>1 and k>1 such that n=m^2 k
    If k is square free, then you are done : b=mē and a=k
    If k is not square free, then there exists integers p>1 and q>1 such that k=p^2 q
    So n=(mp)^2 q

    If q is square free...
    If q is not square free...

    And so on...
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  4. #4
    Member TheMasterMind's Avatar
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    Quote Originally Posted by Moo View Post
    It's the exact definition of square free

    If n is square-free, then a=n and b=1
    If n is a square, then a=1 and b=n

    If n is not square free, nor a square, it means that there exists integers m>1 and k>1 such that n=m^2 k
    If k is square free, then you are done : b=mē and a=k
    If k is not square free, then there exists integers p>1 and q>1 such that k=p^2 q
    So n=(mp)^2 q

    If q is square free...
    If q is not square free...

    And so on...
    Very nice, I was thinking of something along those lines but wanted to check the definition first, make sure my memory is not deceiving me!
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