Find the remainder from the division of $3^{2017}-1$ into $3^{403}-1$











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Here is an interesting problem:



Find the remainder from the division of $3^{2017}-1$ into $3^{403}-1$






elementary-number-theory arithmetic divisibility







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  • Since you are a fairly new user, I would first of all like to welcome you to our stack exchange! Secondly, I suggest that you put some effort, thoughts or even any knowledge you have over the subject of your questions. This way, your posts will be way better well taken !
    – Rebellos
    Nov 19 at 19:23










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    Nov 19 at 20:32















up vote
0
down vote

favorite












Here is an interesting problem:



Find the remainder from the division of $3^{2017}-1$ into $3^{403}-1$






elementary-number-theory arithmetic divisibility







share|cite|improve this question
























  • Since you are a fairly new user, I would first of all like to welcome you to our stack exchange! Secondly, I suggest that you put some effort, thoughts or even any knowledge you have over the subject of your questions. This way, your posts will be way better well taken !
    – Rebellos
    Nov 19 at 19:23










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    Nov 19 at 20:32













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Here is an interesting problem:



Find the remainder from the division of $3^{2017}-1$ into $3^{403}-1$






elementary-number-theory arithmetic divisibility







share|cite|improve this question















Here is an interesting problem:



Find the remainder from the division of $3^{2017}-1$ into $3^{403}-1$






elementary-number-theory arithmetic divisibility




elementary-number-theory arithmetic divisibility






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 20 at 19:19









Martin Sleziak

44.5k7115268




44.5k7115268










asked Nov 19 at 18:31









ten1o

1335




1335












  • Since you are a fairly new user, I would first of all like to welcome you to our stack exchange! Secondly, I suggest that you put some effort, thoughts or even any knowledge you have over the subject of your questions. This way, your posts will be way better well taken !
    – Rebellos
    Nov 19 at 19:23










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    Nov 19 at 20:32


















  • Since you are a fairly new user, I would first of all like to welcome you to our stack exchange! Secondly, I suggest that you put some effort, thoughts or even any knowledge you have over the subject of your questions. This way, your posts will be way better well taken !
    – Rebellos
    Nov 19 at 19:23










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    Nov 19 at 20:32
















Since you are a fairly new user, I would first of all like to welcome you to our stack exchange! Secondly, I suggest that you put some effort, thoughts or even any knowledge you have over the subject of your questions. This way, your posts will be way better well taken !
– Rebellos
Nov 19 at 19:23




Since you are a fairly new user, I would first of all like to welcome you to our stack exchange! Secondly, I suggest that you put some effort, thoughts or even any knowledge you have over the subject of your questions. This way, your posts will be way better well taken !
– Rebellos
Nov 19 at 19:23












Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
– Bill Dubuque
Nov 19 at 20:32




Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
– Bill Dubuque
Nov 19 at 20:32










3 Answers
3






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up vote
4
down vote













Hint :



$$3^{2017} = Big(3^{403}Big)^5 cdot 3^2$$






share|cite|improve this answer






























    up vote
    3
    down vote













    $$3^{403}equiv 1pmod{3^{403}-1}$$
    Raise to the power 5. You get,



    $$3^{2015}equiv 1pmod{3^{403}-1}$$



    $$3^{2017}equiv 9pmod{3^{403}-1}$$



    $$3^{2017}-1equiv 8pmod{3^{403}-1}$$



    So the remainder is 8.






    share|cite|improve this answer




























      up vote
      2
      down vote













      If $n=3^{403}$, you are dividing $9n^5-1$ by $n-1$.



      But $9n^5-1=9(n^5-1)+8$ and the remainder is $8$.






      share|cite|improve this answer





















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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes








        up vote
        4
        down vote













        Hint :



        $$3^{2017} = Big(3^{403}Big)^5 cdot 3^2$$






        share|cite|improve this answer



























          up vote
          4
          down vote













          Hint :



          $$3^{2017} = Big(3^{403}Big)^5 cdot 3^2$$






          share|cite|improve this answer

























            up vote
            4
            down vote










            up vote
            4
            down vote









            Hint :



            $$3^{2017} = Big(3^{403}Big)^5 cdot 3^2$$






            share|cite|improve this answer














            Hint :



            $$3^{2017} = Big(3^{403}Big)^5 cdot 3^2$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Nov 19 at 18:41









            J.G.

            19.8k21932




            19.8k21932










            answered Nov 19 at 18:39









            Rebellos

            12.8k21041




            12.8k21041






















                up vote
                3
                down vote













                $$3^{403}equiv 1pmod{3^{403}-1}$$
                Raise to the power 5. You get,



                $$3^{2015}equiv 1pmod{3^{403}-1}$$



                $$3^{2017}equiv 9pmod{3^{403}-1}$$



                $$3^{2017}-1equiv 8pmod{3^{403}-1}$$



                So the remainder is 8.






                share|cite|improve this answer

























                  up vote
                  3
                  down vote













                  $$3^{403}equiv 1pmod{3^{403}-1}$$
                  Raise to the power 5. You get,



                  $$3^{2015}equiv 1pmod{3^{403}-1}$$



                  $$3^{2017}equiv 9pmod{3^{403}-1}$$



                  $$3^{2017}-1equiv 8pmod{3^{403}-1}$$



                  So the remainder is 8.






                  share|cite|improve this answer























                    up vote
                    3
                    down vote










                    up vote
                    3
                    down vote









                    $$3^{403}equiv 1pmod{3^{403}-1}$$
                    Raise to the power 5. You get,



                    $$3^{2015}equiv 1pmod{3^{403}-1}$$



                    $$3^{2017}equiv 9pmod{3^{403}-1}$$



                    $$3^{2017}-1equiv 8pmod{3^{403}-1}$$



                    So the remainder is 8.






                    share|cite|improve this answer












                    $$3^{403}equiv 1pmod{3^{403}-1}$$
                    Raise to the power 5. You get,



                    $$3^{2015}equiv 1pmod{3^{403}-1}$$



                    $$3^{2017}equiv 9pmod{3^{403}-1}$$



                    $$3^{2017}-1equiv 8pmod{3^{403}-1}$$



                    So the remainder is 8.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Nov 19 at 18:46







                    user612946





























                        up vote
                        2
                        down vote













                        If $n=3^{403}$, you are dividing $9n^5-1$ by $n-1$.



                        But $9n^5-1=9(n^5-1)+8$ and the remainder is $8$.






                        share|cite|improve this answer

























                          up vote
                          2
                          down vote













                          If $n=3^{403}$, you are dividing $9n^5-1$ by $n-1$.



                          But $9n^5-1=9(n^5-1)+8$ and the remainder is $8$.






                          share|cite|improve this answer























                            up vote
                            2
                            down vote










                            up vote
                            2
                            down vote









                            If $n=3^{403}$, you are dividing $9n^5-1$ by $n-1$.



                            But $9n^5-1=9(n^5-1)+8$ and the remainder is $8$.






                            share|cite|improve this answer












                            If $n=3^{403}$, you are dividing $9n^5-1$ by $n-1$.



                            But $9n^5-1=9(n^5-1)+8$ and the remainder is $8$.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Nov 19 at 18:47









                            Yves Daoust

                            122k668218




                            122k668218






























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