Every positive power of 5 appears in the last digits of bigger power of 5












4














Problem. Show that for every positive integer $n$, there is an integer $N > n$ such that the number $5^n$ appears as the last few digits $5^N$. For example, if $n = 3$, we have $5^3 = 125$ and $5^5 = 3125$, so $N = 5$ would satisfy.



Please give hints towards the right direction and not the full solutions. Thanks!!






number-theory contest-math







share|cite|improve this question







New contributor




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

























    4














    Problem. Show that for every positive integer $n$, there is an integer $N > n$ such that the number $5^n$ appears as the last few digits $5^N$. For example, if $n = 3$, we have $5^3 = 125$ and $5^5 = 3125$, so $N = 5$ would satisfy.



    Please give hints towards the right direction and not the full solutions. Thanks!!






    number-theory contest-math







    share|cite|improve this question







    New contributor




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























      4












      4








      4


      1





      Problem. Show that for every positive integer $n$, there is an integer $N > n$ such that the number $5^n$ appears as the last few digits $5^N$. For example, if $n = 3$, we have $5^3 = 125$ and $5^5 = 3125$, so $N = 5$ would satisfy.



      Please give hints towards the right direction and not the full solutions. Thanks!!






      number-theory contest-math







      share|cite|improve this question







      New contributor




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











      Problem. Show that for every positive integer $n$, there is an integer $N > n$ such that the number $5^n$ appears as the last few digits $5^N$. For example, if $n = 3$, we have $5^3 = 125$ and $5^5 = 3125$, so $N = 5$ would satisfy.



      Please give hints towards the right direction and not the full solutions. Thanks!!






      number-theory contest-math




      number-theory contest-math






      share|cite|improve this question







      New contributor




      BrianH 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 question







      New contributor




      BrianH 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 question




      share|cite|improve this question






      New contributor




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









      asked 1 hour ago









      BrianH

      386




      386




      New contributor




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





      New contributor





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






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






















          2 Answers
          2






          active

          oldest

          votes


















          4














          Fix $n$.
          The cases $n le 3$ can be handled directly.
          We now assume $n > 3$.



          Let $m = lceil n log_{10} 5 rceil$ be the smallest integer such that $10^m > 5^n$. For $n > 3$, we have $m < n$.



          You want to find $N$ such that $5^N = k 10^m + 5^n$ for some integer $k$. Dividing both sides by $5^n$ yields $$5^{N-n} - 1 = k 2^m/ 5^{n-m}.$$



          Thus if you show you can find a large integer $q$ such that $5^q - 1$ is divisible by $2^m$ then you can choose $N$ and $k$ appropriately to conclude the proof.





          Base case: For $m=2$ we have $5^1 - 1$ divisible by $2^m$. Inductive step: if $5^q-1$ is divisible by $2^m$, then $5^{2q}-1 = (5^q-1)(5^q+1)$ is divisible by $2^{m+1}$.







          share|cite|improve this answer





























            2














            We will prove that there exists a $N$ such that $5^n$ and $5^N$ have the same last $n$ digits



            So what we need to do is to find $5^Nequiv 5^npmod{10^n}$



            This can be achieved by setting $N=n+phi (2^n)$



            Since $5^Nequiv 5^npmod{5^n}$ and $5^Nequiv 5^npmod{2^n}$






            share|cite|improve this answer








            New contributor




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


















            • Could you explain to me how you reached the last two lines?
              – BrianH
              59 mins ago











            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',
            autoActivateHeartbeat: false,
            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
            });


            }
            });






            BrianH is a new contributor. Be nice, and check out our Code of Conduct.










            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049126%2fevery-positive-power-of-5-appears-in-the-last-digits-of-bigger-power-of-5%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            2 Answers
            2






            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            4














            Fix $n$.
            The cases $n le 3$ can be handled directly.
            We now assume $n > 3$.



            Let $m = lceil n log_{10} 5 rceil$ be the smallest integer such that $10^m > 5^n$. For $n > 3$, we have $m < n$.



            You want to find $N$ such that $5^N = k 10^m + 5^n$ for some integer $k$. Dividing both sides by $5^n$ yields $$5^{N-n} - 1 = k 2^m/ 5^{n-m}.$$



            Thus if you show you can find a large integer $q$ such that $5^q - 1$ is divisible by $2^m$ then you can choose $N$ and $k$ appropriately to conclude the proof.





            Base case: For $m=2$ we have $5^1 - 1$ divisible by $2^m$. Inductive step: if $5^q-1$ is divisible by $2^m$, then $5^{2q}-1 = (5^q-1)(5^q+1)$ is divisible by $2^{m+1}$.







            share|cite|improve this answer


























              4














              Fix $n$.
              The cases $n le 3$ can be handled directly.
              We now assume $n > 3$.



              Let $m = lceil n log_{10} 5 rceil$ be the smallest integer such that $10^m > 5^n$. For $n > 3$, we have $m < n$.



              You want to find $N$ such that $5^N = k 10^m + 5^n$ for some integer $k$. Dividing both sides by $5^n$ yields $$5^{N-n} - 1 = k 2^m/ 5^{n-m}.$$



              Thus if you show you can find a large integer $q$ such that $5^q - 1$ is divisible by $2^m$ then you can choose $N$ and $k$ appropriately to conclude the proof.





              Base case: For $m=2$ we have $5^1 - 1$ divisible by $2^m$. Inductive step: if $5^q-1$ is divisible by $2^m$, then $5^{2q}-1 = (5^q-1)(5^q+1)$ is divisible by $2^{m+1}$.







              share|cite|improve this answer
























                4












                4








                4






                Fix $n$.
                The cases $n le 3$ can be handled directly.
                We now assume $n > 3$.



                Let $m = lceil n log_{10} 5 rceil$ be the smallest integer such that $10^m > 5^n$. For $n > 3$, we have $m < n$.



                You want to find $N$ such that $5^N = k 10^m + 5^n$ for some integer $k$. Dividing both sides by $5^n$ yields $$5^{N-n} - 1 = k 2^m/ 5^{n-m}.$$



                Thus if you show you can find a large integer $q$ such that $5^q - 1$ is divisible by $2^m$ then you can choose $N$ and $k$ appropriately to conclude the proof.





                Base case: For $m=2$ we have $5^1 - 1$ divisible by $2^m$. Inductive step: if $5^q-1$ is divisible by $2^m$, then $5^{2q}-1 = (5^q-1)(5^q+1)$ is divisible by $2^{m+1}$.







                share|cite|improve this answer












                Fix $n$.
                The cases $n le 3$ can be handled directly.
                We now assume $n > 3$.



                Let $m = lceil n log_{10} 5 rceil$ be the smallest integer such that $10^m > 5^n$. For $n > 3$, we have $m < n$.



                You want to find $N$ such that $5^N = k 10^m + 5^n$ for some integer $k$. Dividing both sides by $5^n$ yields $$5^{N-n} - 1 = k 2^m/ 5^{n-m}.$$



                Thus if you show you can find a large integer $q$ such that $5^q - 1$ is divisible by $2^m$ then you can choose $N$ and $k$ appropriately to conclude the proof.





                Base case: For $m=2$ we have $5^1 - 1$ divisible by $2^m$. Inductive step: if $5^q-1$ is divisible by $2^m$, then $5^{2q}-1 = (5^q-1)(5^q+1)$ is divisible by $2^{m+1}$.








                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 1 hour ago









                angryavian

                38.4k23180




                38.4k23180























                    2














                    We will prove that there exists a $N$ such that $5^n$ and $5^N$ have the same last $n$ digits



                    So what we need to do is to find $5^Nequiv 5^npmod{10^n}$



                    This can be achieved by setting $N=n+phi (2^n)$



                    Since $5^Nequiv 5^npmod{5^n}$ and $5^Nequiv 5^npmod{2^n}$






                    share|cite|improve this answer








                    New contributor




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


















                    • Could you explain to me how you reached the last two lines?
                      – BrianH
                      59 mins ago
















                    2














                    We will prove that there exists a $N$ such that $5^n$ and $5^N$ have the same last $n$ digits



                    So what we need to do is to find $5^Nequiv 5^npmod{10^n}$



                    This can be achieved by setting $N=n+phi (2^n)$



                    Since $5^Nequiv 5^npmod{5^n}$ and $5^Nequiv 5^npmod{2^n}$






                    share|cite|improve this answer








                    New contributor




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


















                    • Could you explain to me how you reached the last two lines?
                      – BrianH
                      59 mins ago














                    2












                    2








                    2






                    We will prove that there exists a $N$ such that $5^n$ and $5^N$ have the same last $n$ digits



                    So what we need to do is to find $5^Nequiv 5^npmod{10^n}$



                    This can be achieved by setting $N=n+phi (2^n)$



                    Since $5^Nequiv 5^npmod{5^n}$ and $5^Nequiv 5^npmod{2^n}$






                    share|cite|improve this answer








                    New contributor




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









                    We will prove that there exists a $N$ such that $5^n$ and $5^N$ have the same last $n$ digits



                    So what we need to do is to find $5^Nequiv 5^npmod{10^n}$



                    This can be achieved by setting $N=n+phi (2^n)$



                    Since $5^Nequiv 5^npmod{5^n}$ and $5^Nequiv 5^npmod{2^n}$







                    share|cite|improve this answer








                    New contributor




                    bangzheng 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




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









                    answered 1 hour ago









                    bangzheng

                    211




                    211




                    New contributor




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





                    New contributor





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






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












                    • Could you explain to me how you reached the last two lines?
                      – BrianH
                      59 mins ago


















                    • Could you explain to me how you reached the last two lines?
                      – BrianH
                      59 mins ago
















                    Could you explain to me how you reached the last two lines?
                    – BrianH
                    59 mins ago




                    Could you explain to me how you reached the last two lines?
                    – BrianH
                    59 mins ago










                    BrianH is a new contributor. Be nice, and check out our Code of Conduct.










                    draft saved

                    draft discarded


















                    BrianH is a new contributor. Be nice, and check out our Code of Conduct.













                    BrianH is a new contributor. Be nice, and check out our Code of Conduct.












                    BrianH is a new contributor. Be nice, and check out our Code of Conduct.
















                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid …



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3049126%2fevery-positive-power-of-5-appears-in-the-last-digits-of-bigger-power-of-5%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

                    404 Error Contact Form 7 ajax form submitting

                    Image
                    0 The contact form does not work, and affects loading after one try. The chrome console shows the error message: POST http://example.com/wp-json/contact-form-7/v1/contact-forms/50/feedback 404 (Not Found) send @ jquery.js?ver=1.12.4:4 ajax @ jquery.js?ver=1.12.4:4 wpcf7.submit @ scripts.js?ver=5.0.3:346 (anonymous) @ scripts.js?ver=5.0.3:53 dispatch @ jquery.js?ver=1.12.4:3 r.handle @ jquery.js?ver=1.12.4:3 How to solve this issue? thanks in advance javascript wordpress contact-form-7 share | improve this question edited Jul 17 '18 at 5:39 Kiran Shahi 4,745 3 16 43 ...
                    Read more

                    How to know if a Active Directory user can login interactively

                    Image
                    1 I would like to know if and how is it possible to know if a AD user can login interactively (on a server) in a domain. I need to know if I can find it out using a LDAP search. Thanks. windows active-directory login user-permissions user-profile share | improve this question edited 1 hour ago yagmoth555 ♦ 11.6k 3 17 42 asked 1 hour ago Luigi Luigi 6 1 New contributor ...
                    Read more

                    Refactoring coordinates for Minecraft Pi buildings written in Python

                    Image
                    0 $begingroup$ I'm fairly new to coding and I wrote this script in Python for a house with a roof in Minecraft Pi. I've found different ways to write the coordinates to make the code base shorter, but I'm wondering if it is possible for me to refactor it any further so I don't have to list each coordinate line by line. import mcpi.minecraft as minecraft import mcpi.block as block mc = minecraft.Minecraft.create() p = mc.player.getTilePos() mc.setBlocks(p.x+1, p.y, p.z+1, #Main structure starting coordinates (DO NOT ALTER) p.x+19, p.y+4, p.z+6, #Structure size/coordinates (Alters cube size) block.STONE_BRICK) mc.setBlocks(p.x+2, p.y+1, p.z+2, #Hollowing out main structure interior p.x+18, p.y+3, p.z+5, block.AIR) mc.setBlocks(p.x, p.y...
                    Read more