a conjecture on the binary operation of multiplication [duplicate]












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  • Prove that: set ${1, 2, 3, …, n - 1}$ is group under multiplication modulo $n$?

    6 answers




Give a conjecture describing the values of $n$ for which all of the nonzero elements of $Z_n = {0, 1, 2, . . . , n − 1}$ have multiplicative inverses.



I am guessing the point of not having $0$ included is so nonprime numbers of $n$ will now have inverses, but I want to make sure this is the case.






binary-operations







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marked as duplicate by JMoravitz, anomaly, Alexander Gruber♦ Nov 30 at 3:06


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • The condition of being nonzero is because 0 never has a multiplicative inverse (in a nontrivial ring). I'm not sure what you mean by "nonprime numbers of $n$ now have inverses."
    – platty
    Nov 30 at 1:18
















0















This question already has an answer here:




  • Prove that: set ${1, 2, 3, …, n - 1}$ is group under multiplication modulo $n$?

    6 answers




Give a conjecture describing the values of $n$ for which all of the nonzero elements of $Z_n = {0, 1, 2, . . . , n − 1}$ have multiplicative inverses.



I am guessing the point of not having $0$ included is so nonprime numbers of $n$ will now have inverses, but I want to make sure this is the case.






binary-operations







share|cite|improve this question













marked as duplicate by JMoravitz, anomaly, Alexander Gruber♦ Nov 30 at 3:06


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • The condition of being nonzero is because 0 never has a multiplicative inverse (in a nontrivial ring). I'm not sure what you mean by "nonprime numbers of $n$ now have inverses."
    – platty
    Nov 30 at 1:18














0












0








0








This question already has an answer here:




  • Prove that: set ${1, 2, 3, …, n - 1}$ is group under multiplication modulo $n$?

    6 answers




Give a conjecture describing the values of $n$ for which all of the nonzero elements of $Z_n = {0, 1, 2, . . . , n − 1}$ have multiplicative inverses.



I am guessing the point of not having $0$ included is so nonprime numbers of $n$ will now have inverses, but I want to make sure this is the case.






binary-operations







share|cite|improve this question














This question already has an answer here:




  • Prove that: set ${1, 2, 3, …, n - 1}$ is group under multiplication modulo $n$?

    6 answers




Give a conjecture describing the values of $n$ for which all of the nonzero elements of $Z_n = {0, 1, 2, . . . , n − 1}$ have multiplicative inverses.



I am guessing the point of not having $0$ included is so nonprime numbers of $n$ will now have inverses, but I want to make sure this is the case.





This question already has an answer here:




  • Prove that: set ${1, 2, 3, …, n - 1}$ is group under multiplication modulo $n$?

    6 answers







binary-operations




binary-operations






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share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 30 at 1:13









George

676




676




marked as duplicate by JMoravitz, anomaly, Alexander Gruber♦ Nov 30 at 3:06


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






marked as duplicate by JMoravitz, anomaly, Alexander Gruber♦ Nov 30 at 3:06


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • The condition of being nonzero is because 0 never has a multiplicative inverse (in a nontrivial ring). I'm not sure what you mean by "nonprime numbers of $n$ now have inverses."
    – platty
    Nov 30 at 1:18


















  • The condition of being nonzero is because 0 never has a multiplicative inverse (in a nontrivial ring). I'm not sure what you mean by "nonprime numbers of $n$ now have inverses."
    – platty
    Nov 30 at 1:18
















The condition of being nonzero is because 0 never has a multiplicative inverse (in a nontrivial ring). I'm not sure what you mean by "nonprime numbers of $n$ now have inverses."
– platty
Nov 30 at 1:18




The condition of being nonzero is because 0 never has a multiplicative inverse (in a nontrivial ring). I'm not sure what you mean by "nonprime numbers of $n$ now have inverses."
– platty
Nov 30 at 1:18










1 Answer
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It is not necessarily the case that all nonprime numbers less than $n$ have inverses.



Hint: $rin{0,1,2,dotsc, n-1}$ has a multiplicative inverse if there is some $s$ such that $rs-1$ is divisible by $n$, i.e., such that $rs-1 = kn$, or $rs-kn = 1$. What does that tell you about $r$ and $n$?






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    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0














    It is not necessarily the case that all nonprime numbers less than $n$ have inverses.



    Hint: $rin{0,1,2,dotsc, n-1}$ has a multiplicative inverse if there is some $s$ such that $rs-1$ is divisible by $n$, i.e., such that $rs-1 = kn$, or $rs-kn = 1$. What does that tell you about $r$ and $n$?






    share|cite|improve this answer


























      0














      It is not necessarily the case that all nonprime numbers less than $n$ have inverses.



      Hint: $rin{0,1,2,dotsc, n-1}$ has a multiplicative inverse if there is some $s$ such that $rs-1$ is divisible by $n$, i.e., such that $rs-1 = kn$, or $rs-kn = 1$. What does that tell you about $r$ and $n$?






      share|cite|improve this answer
























        0












        0








        0






        It is not necessarily the case that all nonprime numbers less than $n$ have inverses.



        Hint: $rin{0,1,2,dotsc, n-1}$ has a multiplicative inverse if there is some $s$ such that $rs-1$ is divisible by $n$, i.e., such that $rs-1 = kn$, or $rs-kn = 1$. What does that tell you about $r$ and $n$?






        share|cite|improve this answer












        It is not necessarily the case that all nonprime numbers less than $n$ have inverses.



        Hint: $rin{0,1,2,dotsc, n-1}$ has a multiplicative inverse if there is some $s$ such that $rs-1$ is divisible by $n$, i.e., such that $rs-1 = kn$, or $rs-kn = 1$. What does that tell you about $r$ and $n$?







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 30 at 1:18









        rogerl

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        17.4k22746















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