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











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
    2 days ago










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    2 days ago















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
    2 days ago










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    2 days ago













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 yesterday









Martin Sleziak

44.3k7115266




44.3k7115266










asked 2 days ago









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
    2 days ago










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    2 days ago


















  • 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
    2 days ago










  • Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
    – Bill Dubuque
    2 days ago
















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
2 days ago




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
2 days ago












Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
– Bill Dubuque
2 days ago




Hint $bmod n-1!: nequiv 1,Rightarrow, f(n)equiv f(1) $ for any polynomial $f(x)$ with integer coefs
– Bill Dubuque
2 days ago










3 Answers
3






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
    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








    New contributor




    Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.

























      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





















        Your Answer





        StackExchange.ifUsing("editor", function () {
        return StackExchange.using("mathjaxEditing", function () {
        StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
        StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
        });
        });
        }, "mathjax-editing");

        StackExchange.ready(function() {
        var channelOptions = {
        tags: "".split(" "),
        id: "69"
        };
        initTagRenderer("".split(" "), "".split(" "), channelOptions);

        StackExchange.using("externalEditor", function() {
        // Have to fire editor after snippets, if snippets enabled
        if (StackExchange.settings.snippets.snippetsEnabled) {
        StackExchange.using("snippets", function() {
        createEditor();
        });
        }
        else {
        createEditor();
        }
        });

        function createEditor() {
        StackExchange.prepareEditor({
        heartbeatType: 'answer',
        convertImagesToLinks: true,
        noModals: true,
        showLowRepImageUploadWarning: true,
        reputationToPostImages: 10,
        bindNavPrevention: true,
        postfix: "",
        imageUploader: {
        brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
        contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
        allowUrls: true
        },
        noCode: true, onDemand: true,
        discardSelector: ".discard-answer"
        ,immediatelyShowMarkdownHelp:true
        });


        }
        });














         

        draft saved


        draft discarded


















        StackExchange.ready(
        function () {
        StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005315%2ffind-the-remainder-from-the-division-of-32017-1-into-3403-1%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        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 2 days ago









            J.G.

            18.4k21932




            18.4k21932










            answered 2 days ago









            Rebellos

            11.5k21040




            11.5k21040






















                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








                New contributor




                Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                Check out our Code of Conduct.






















                  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








                  New contributor




                  Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                  Check out our Code of Conduct.




















                    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








                    New contributor




                    Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









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








                    New contributor




                    Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    share|cite|improve this answer



                    share|cite|improve this answer






                    New contributor




                    Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.









                    answered 2 days ago









                    Samurai

                    987




                    987




                    New contributor




                    Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.





                    New contributor





                    Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.






                    Samurai is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
                    Check out our Code of Conduct.






















                        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 2 days ago









                            Yves Daoust

                            121k668218




                            121k668218






























                                 

                                draft saved


                                draft discarded



















































                                 


                                draft saved


                                draft discarded














                                StackExchange.ready(
                                function () {
                                StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005315%2ffind-the-remainder-from-the-division-of-32017-1-into-3403-1%23new-answer', 'question_page');
                                }
                                );

                                Post as a guest















                                Required, but never shown





















































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown

































                                Required, but never shown














                                Required, but never shown












                                Required, but never shown







                                Required, but never shown







                                Popular posts from this blog

                                MongoDB - Not Authorized To Execute Command

                                Image
                                .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I have successfully enabled authorization on MongoDB and I have created an account on the admin database and then I created an account for my database called test. The following connection string to connect to my test database works successfully: mongo --host 192.168.17.52 --port 27017 -u user1 -p password --authenticationDatabase test Only problem I have now is, I cannot execute commands such as: show dbs. I get the following error when I try to do so: "errmsg" : "not authorized on admin to execute command { listDatabases: 1.0, lsid: { id: UUID("a1d5bc0d-bc58-485e-b232-270758a89455") }, $db: "admin...
                                Read more

                                How to fix TextFormField cause rebuild widget in Flutter

                                Image
                                .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty{ height:90px;width:728px;box-sizing:border-box; } 0 I've got the same issue at #9280. I tried all solutions from that topic with/without GlobalKey for Scaffold and Form but the problem wasn't fixed. I tried to build a authentication form with login and signup. But the keyboard doesn't show when I tap the TextFormField at the first time. Then I try to input some letter from physical keyboard, the widget is rebuilt. Like this: login screen Nothing to be log when I run flutter run --verbose, so I logged manually. Every tapping on field, the widget called dispose -> init -> build. [√] Flutter (Channel beta, v1.0.0, on Microsoft Windows [Version 10.0.17763.195], ...
                                Read more

                                in spring boot 2.1 many test slices are not allowed anymore due to multiple @BootstrapWith

                                Image
                                1 I tried to upgrade a yummy sandwich made of two test slices (@JsonTest and @JdbcTest in my case, crunchy test code in between) adding spring boot 2.1 flavour to it. But it seems it was not much of a success. I cannot annotate my tests with many @...Test since they are now each bringing their own XxxTestContextBootstrapper. It used to work when they all used same SpringBootTestContextBootstrapper. @RunWith(SpringRunner.class) @JdbcTest @JsonTest public class Test { @Test public void test() { System.out.printn("Hello, World !"); } } The error I get from BootstrapUtils is illegalStateException : Configuration error: found multiple declarations of @BootstrapWith for test class I understand I might be doing something wrong here but is there an easy way I could load both Json and Jdbc contex...
                                Read more