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Math Help - Integers

  1. #1
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    Integers

    Find an integer n > 20 so that the equation 3x = b always have a unigue solution in Zn regardless of the value b


    Not to sure what Zn means
    Can the value of b be anything?

    Please help
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  2. #2
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    Quote Originally Posted by mathlg
    Find an integer n > 20 so that the equation 3x = b always have a unigue solution in Zn regardless of the value b


    Not to sure what Zn means
    Can the value of b be anything?

    Please help
    By,
    \mathbb{Z}_n we mean the group of positive integers added modulo n.

    I believe the theorem goes that,
    ax=b has a solution in \mathbb{Z}_n when, \gcd(b,n)=1.
    For your problem you need to find all the integers relatively prime to 20 and less. This is called the 'phi-function'. Thus, you need to find the smalles n>20 such as all the previous integers are relatively prime to it. Meaning a prime number, the smallest after 20 is 23.
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  3. #3
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    So you are saying that since n > 20 the next smallest prime number is 23. That would be the integer im looking for in the problem? I get confused when it says regardless of the value b.
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  4. #4
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    Quote Originally Posted by mathlg
    So you are saying that since n > 20 the next smallest prime number is 23. That would be the integer im looking for in the problem? I get confused when it says regardless of the value b.
    Yes
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