How is this property called for mod?












7












$begingroup$


We have a name for the property of integers to be $0$ or $1$ $mathrm{mod} 2$ - parity.



Is there any similar name for the remainder for any other base? Like a generalization of parity? Could I use parity in a broader sense, just to name the remainder $mathrm{mod} n$?






elementary-number-theory soft-question integers







share|cite|improve this question









$endgroup$








  • 9




    $begingroup$
    How about congruence?
    $endgroup$
    – Umberto P.
    Feb 25 at 11:37










  • $begingroup$
    So should I say (variable)'s congruence class or value?
    $endgroup$
    – dEmigOd
    Feb 25 at 11:43










  • $begingroup$
    I had the same question and thought about terminology for $n = 3$. A friend of mine came up with a very neat suggestion: flat, short and long. I don't know if there is any other generalization, but I would very much like to hear people talking about remainders mod 3 like this. I guess the reason why this has a name for $n=2$ is that it appears commonly in everyday life as opposed to most of the other numbers. When talking about time, the 12 is dropped at all, but this is somewhat different maybe
    $endgroup$
    – kesa
    Feb 25 at 15:44


















7












$begingroup$


We have a name for the property of integers to be $0$ or $1$ $mathrm{mod} 2$ - parity.



Is there any similar name for the remainder for any other base? Like a generalization of parity? Could I use parity in a broader sense, just to name the remainder $mathrm{mod} n$?






elementary-number-theory soft-question integers







share|cite|improve this question









$endgroup$








  • 9




    $begingroup$
    How about congruence?
    $endgroup$
    – Umberto P.
    Feb 25 at 11:37










  • $begingroup$
    So should I say (variable)'s congruence class or value?
    $endgroup$
    – dEmigOd
    Feb 25 at 11:43










  • $begingroup$
    I had the same question and thought about terminology for $n = 3$. A friend of mine came up with a very neat suggestion: flat, short and long. I don't know if there is any other generalization, but I would very much like to hear people talking about remainders mod 3 like this. I guess the reason why this has a name for $n=2$ is that it appears commonly in everyday life as opposed to most of the other numbers. When talking about time, the 12 is dropped at all, but this is somewhat different maybe
    $endgroup$
    – kesa
    Feb 25 at 15:44
















7












7








7





$begingroup$


We have a name for the property of integers to be $0$ or $1$ $mathrm{mod} 2$ - parity.



Is there any similar name for the remainder for any other base? Like a generalization of parity? Could I use parity in a broader sense, just to name the remainder $mathrm{mod} n$?






elementary-number-theory soft-question integers







share|cite|improve this question









$endgroup$




We have a name for the property of integers to be $0$ or $1$ $mathrm{mod} 2$ - parity.



Is there any similar name for the remainder for any other base? Like a generalization of parity? Could I use parity in a broader sense, just to name the remainder $mathrm{mod} n$?






elementary-number-theory soft-question integers




elementary-number-theory soft-question integers






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Feb 25 at 11:34









dEmigOddEmigOd

1,5411612




1,5411612








  • 9




    $begingroup$
    How about congruence?
    $endgroup$
    – Umberto P.
    Feb 25 at 11:37










  • $begingroup$
    So should I say (variable)'s congruence class or value?
    $endgroup$
    – dEmigOd
    Feb 25 at 11:43










  • $begingroup$
    I had the same question and thought about terminology for $n = 3$. A friend of mine came up with a very neat suggestion: flat, short and long. I don't know if there is any other generalization, but I would very much like to hear people talking about remainders mod 3 like this. I guess the reason why this has a name for $n=2$ is that it appears commonly in everyday life as opposed to most of the other numbers. When talking about time, the 12 is dropped at all, but this is somewhat different maybe
    $endgroup$
    – kesa
    Feb 25 at 15:44
















  • 9




    $begingroup$
    How about congruence?
    $endgroup$
    – Umberto P.
    Feb 25 at 11:37










  • $begingroup$
    So should I say (variable)'s congruence class or value?
    $endgroup$
    – dEmigOd
    Feb 25 at 11:43










  • $begingroup$
    I had the same question and thought about terminology for $n = 3$. A friend of mine came up with a very neat suggestion: flat, short and long. I don't know if there is any other generalization, but I would very much like to hear people talking about remainders mod 3 like this. I guess the reason why this has a name for $n=2$ is that it appears commonly in everyday life as opposed to most of the other numbers. When talking about time, the 12 is dropped at all, but this is somewhat different maybe
    $endgroup$
    – kesa
    Feb 25 at 15:44










9




9




$begingroup$
How about congruence?
$endgroup$
– Umberto P.
Feb 25 at 11:37




$begingroup$
How about congruence?
$endgroup$
– Umberto P.
Feb 25 at 11:37












$begingroup$
So should I say (variable)'s congruence class or value?
$endgroup$
– dEmigOd
Feb 25 at 11:43




$begingroup$
So should I say (variable)'s congruence class or value?
$endgroup$
– dEmigOd
Feb 25 at 11:43












$begingroup$
I had the same question and thought about terminology for $n = 3$. A friend of mine came up with a very neat suggestion: flat, short and long. I don't know if there is any other generalization, but I would very much like to hear people talking about remainders mod 3 like this. I guess the reason why this has a name for $n=2$ is that it appears commonly in everyday life as opposed to most of the other numbers. When talking about time, the 12 is dropped at all, but this is somewhat different maybe
$endgroup$
– kesa
Feb 25 at 15:44






$begingroup$
I had the same question and thought about terminology for $n = 3$. A friend of mine came up with a very neat suggestion: flat, short and long. I don't know if there is any other generalization, but I would very much like to hear people talking about remainders mod 3 like this. I guess the reason why this has a name for $n=2$ is that it appears commonly in everyday life as opposed to most of the other numbers. When talking about time, the 12 is dropped at all, but this is somewhat different maybe
$endgroup$
– kesa
Feb 25 at 15:44












4 Answers
4






active

oldest

votes


















9












$begingroup$

Simply say "congruent to $a$ modulo $m$" to read "$equiv a pmod{m}$".






share|cite|improve this answer









$endgroup$





















    16












    $begingroup$

    Actually there is a standard name: residue.




    There are $5$ residues modulo $5$, namely $0,1,2,3,4$.



    Every prime greater than $3$ falls into only $2$ residue-classes modulo $6$.







    share|cite|improve this answer









    $endgroup$





















      2












      $begingroup$

      In a broader sense, when you are dealing with congruences you are dealing with an equivalence relation and its equivalence classes.



      The basic idea of an equivalence relation is to collect elements that are different but behave in the same manner with respect to some property of interest.



      In modular arithmetic this property is having the same rest when divided by a prescribed integer



      If $a=bbmod m$ or $aequiv_m b$ you essentially say that $a$ and $b$ are in the same equivalence class with respect to the equivalence relation $equiv_m$.



      So you could say that "$a$ and $b$ are in the same equivalence class when we look at the remainder upon dividing by $m$", which definitely longer and more cumbersome to say "$a$ congruent to $b$ modulo $m$" but nonetheless another way to express the same concept.






      share|cite|improve this answer











      $endgroup$





















        -1












        $begingroup$

        Either AlessioDV's good answer, or you could say: "of the form $nq+r$". as a representation of $equiv r pmod n$ for example a number $N$ with $$Nequiv 1pmod 3$$ has property $N=3q+1$.






        share|cite|improve this answer









        $endgroup$













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          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          9












          $begingroup$

          Simply say "congruent to $a$ modulo $m$" to read "$equiv a pmod{m}$".






          share|cite|improve this answer









          $endgroup$


















            9












            $begingroup$

            Simply say "congruent to $a$ modulo $m$" to read "$equiv a pmod{m}$".






            share|cite|improve this answer









            $endgroup$
















              9












              9








              9





              $begingroup$

              Simply say "congruent to $a$ modulo $m$" to read "$equiv a pmod{m}$".






              share|cite|improve this answer









              $endgroup$



              Simply say "congruent to $a$ modulo $m$" to read "$equiv a pmod{m}$".







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Feb 25 at 11:39









              AlessioDVAlessioDV

              667114




              667114























                  16












                  $begingroup$

                  Actually there is a standard name: residue.




                  There are $5$ residues modulo $5$, namely $0,1,2,3,4$.



                  Every prime greater than $3$ falls into only $2$ residue-classes modulo $6$.







                  share|cite|improve this answer









                  $endgroup$


















                    16












                    $begingroup$

                    Actually there is a standard name: residue.




                    There are $5$ residues modulo $5$, namely $0,1,2,3,4$.



                    Every prime greater than $3$ falls into only $2$ residue-classes modulo $6$.







                    share|cite|improve this answer









                    $endgroup$
















                      16












                      16








                      16





                      $begingroup$

                      Actually there is a standard name: residue.




                      There are $5$ residues modulo $5$, namely $0,1,2,3,4$.



                      Every prime greater than $3$ falls into only $2$ residue-classes modulo $6$.







                      share|cite|improve this answer









                      $endgroup$



                      Actually there is a standard name: residue.




                      There are $5$ residues modulo $5$, namely $0,1,2,3,4$.



                      Every prime greater than $3$ falls into only $2$ residue-classes modulo $6$.








                      share|cite|improve this answer












                      share|cite|improve this answer



                      share|cite|improve this answer










                      answered Feb 25 at 15:56









                      user21820user21820

                      39.3k543154




                      39.3k543154























                          2












                          $begingroup$

                          In a broader sense, when you are dealing with congruences you are dealing with an equivalence relation and its equivalence classes.



                          The basic idea of an equivalence relation is to collect elements that are different but behave in the same manner with respect to some property of interest.



                          In modular arithmetic this property is having the same rest when divided by a prescribed integer



                          If $a=bbmod m$ or $aequiv_m b$ you essentially say that $a$ and $b$ are in the same equivalence class with respect to the equivalence relation $equiv_m$.



                          So you could say that "$a$ and $b$ are in the same equivalence class when we look at the remainder upon dividing by $m$", which definitely longer and more cumbersome to say "$a$ congruent to $b$ modulo $m$" but nonetheless another way to express the same concept.






                          share|cite|improve this answer











                          $endgroup$


















                            2












                            $begingroup$

                            In a broader sense, when you are dealing with congruences you are dealing with an equivalence relation and its equivalence classes.



                            The basic idea of an equivalence relation is to collect elements that are different but behave in the same manner with respect to some property of interest.



                            In modular arithmetic this property is having the same rest when divided by a prescribed integer



                            If $a=bbmod m$ or $aequiv_m b$ you essentially say that $a$ and $b$ are in the same equivalence class with respect to the equivalence relation $equiv_m$.



                            So you could say that "$a$ and $b$ are in the same equivalence class when we look at the remainder upon dividing by $m$", which definitely longer and more cumbersome to say "$a$ congruent to $b$ modulo $m$" but nonetheless another way to express the same concept.






                            share|cite|improve this answer











                            $endgroup$
















                              2












                              2








                              2





                              $begingroup$

                              In a broader sense, when you are dealing with congruences you are dealing with an equivalence relation and its equivalence classes.



                              The basic idea of an equivalence relation is to collect elements that are different but behave in the same manner with respect to some property of interest.



                              In modular arithmetic this property is having the same rest when divided by a prescribed integer



                              If $a=bbmod m$ or $aequiv_m b$ you essentially say that $a$ and $b$ are in the same equivalence class with respect to the equivalence relation $equiv_m$.



                              So you could say that "$a$ and $b$ are in the same equivalence class when we look at the remainder upon dividing by $m$", which definitely longer and more cumbersome to say "$a$ congruent to $b$ modulo $m$" but nonetheless another way to express the same concept.






                              share|cite|improve this answer











                              $endgroup$



                              In a broader sense, when you are dealing with congruences you are dealing with an equivalence relation and its equivalence classes.



                              The basic idea of an equivalence relation is to collect elements that are different but behave in the same manner with respect to some property of interest.



                              In modular arithmetic this property is having the same rest when divided by a prescribed integer



                              If $a=bbmod m$ or $aequiv_m b$ you essentially say that $a$ and $b$ are in the same equivalence class with respect to the equivalence relation $equiv_m$.



                              So you could say that "$a$ and $b$ are in the same equivalence class when we look at the remainder upon dividing by $m$", which definitely longer and more cumbersome to say "$a$ congruent to $b$ modulo $m$" but nonetheless another way to express the same concept.







                              share|cite|improve this answer














                              share|cite|improve this answer



                              share|cite|improve this answer








                              edited Feb 25 at 14:53









                              J. W. Tanner

                              2,6861217




                              2,6861217










                              answered Feb 25 at 11:52









                              Vinyl_coat_jawaVinyl_coat_jawa

                              3,20511233




                              3,20511233























                                  -1












                                  $begingroup$

                                  Either AlessioDV's good answer, or you could say: "of the form $nq+r$". as a representation of $equiv r pmod n$ for example a number $N$ with $$Nequiv 1pmod 3$$ has property $N=3q+1$.






                                  share|cite|improve this answer









                                  $endgroup$


















                                    -1












                                    $begingroup$

                                    Either AlessioDV's good answer, or you could say: "of the form $nq+r$". as a representation of $equiv r pmod n$ for example a number $N$ with $$Nequiv 1pmod 3$$ has property $N=3q+1$.






                                    share|cite|improve this answer









                                    $endgroup$
















                                      -1












                                      -1








                                      -1





                                      $begingroup$

                                      Either AlessioDV's good answer, or you could say: "of the form $nq+r$". as a representation of $equiv r pmod n$ for example a number $N$ with $$Nequiv 1pmod 3$$ has property $N=3q+1$.






                                      share|cite|improve this answer









                                      $endgroup$



                                      Either AlessioDV's good answer, or you could say: "of the form $nq+r$". as a representation of $equiv r pmod n$ for example a number $N$ with $$Nequiv 1pmod 3$$ has property $N=3q+1$.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Feb 25 at 11:49









                                      Rhys HughesRhys Hughes

                                      6,9571530




                                      6,9571530






























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