What is the smallest number n> 5 so that 5 ^ n ends with “3125”?












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What is the smallest number n> 5 so that 5 ^ n ends with "3125"?



What other examples are there?






calculus







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  • 1




    $begingroup$
    What is your take on this?
    $endgroup$
    – ADITYA PRAKASH
    2 hours ago






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    2 hours ago






  • 3




    $begingroup$
    Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
    $endgroup$
    – fleablood
    2 hours ago
















1












$begingroup$


What is the smallest number n> 5 so that 5 ^ n ends with "3125"?



What other examples are there?






calculus







share|cite|improve this question







New contributor




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







$endgroup$








  • 1




    $begingroup$
    What is your take on this?
    $endgroup$
    – ADITYA PRAKASH
    2 hours ago






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    2 hours ago






  • 3




    $begingroup$
    Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
    $endgroup$
    – fleablood
    2 hours ago














1












1








1





$begingroup$


What is the smallest number n> 5 so that 5 ^ n ends with "3125"?



What other examples are there?






calculus







share|cite|improve this question







New contributor




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







$endgroup$




What is the smallest number n> 5 so that 5 ^ n ends with "3125"?



What other examples are there?






calculus




calculus






share|cite|improve this question







New contributor




Catherine Cooper 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




Catherine Cooper 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




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









asked 2 hours ago









Catherine Cooper Catherine Cooper

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New contributor




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





New contributor





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






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








  • 1




    $begingroup$
    What is your take on this?
    $endgroup$
    – ADITYA PRAKASH
    2 hours ago






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    2 hours ago






  • 3




    $begingroup$
    Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
    $endgroup$
    – fleablood
    2 hours ago














  • 1




    $begingroup$
    What is your take on this?
    $endgroup$
    – ADITYA PRAKASH
    2 hours ago






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    2 hours ago






  • 3




    $begingroup$
    Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
    $endgroup$
    – fleablood
    2 hours ago








1




1




$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
2 hours ago




$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
2 hours ago




1




1




$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
2 hours ago




$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
2 hours ago




3




3




$begingroup$
Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
$endgroup$
– fleablood
2 hours ago




$begingroup$
Why not just do it? It's not $1$ because $5^1=5$. It's not $2$ because $5^2 = 25$. What's to keep you from just continuing?
$endgroup$
– fleablood
2 hours ago










4 Answers
4






active

oldest

votes


















4












$begingroup$


So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.




Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$



Conclusion




All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.







share|cite|improve this answer









$endgroup$





















    2












    $begingroup$

    Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?






    share|cite|improve this answer









    $endgroup$





















      0












      $begingroup$

      Well
      $$5^9=1953125$$
      so the answer is $9$. In fact
      $$5^nequiv 5^{n-4} mod{10^4}$$
      For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        Why not $5^5 = 3125$.
        $endgroup$
        – fleablood
        2 hours ago






      • 1




        $begingroup$
        The question states that $ngt5$
        $endgroup$
        – Peter Foreman
        2 hours ago



















      0












      $begingroup$

      Hint $, 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}).,$ Now recall $, begin{align} 5, &equiv 1!pmod{! color{#c00}4} \ Rightarrow 5^{largecolor{#c00} 4}!&equiv 1^{largecolor{#c00} 4}!!!! pmod{!color{#c00}4^{large 2}}end{align}$






      share|cite|improve this answer











      $endgroup$













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        4 Answers
        4






        active

        oldest

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        4 Answers
        4






        active

        oldest

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        active

        oldest

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        active

        oldest

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        4












        $begingroup$


        So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.




        Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$



        Conclusion




        All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.







        share|cite|improve this answer









        $endgroup$


















          4












          $begingroup$


          So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.




          Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$



          Conclusion




          All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.







          share|cite|improve this answer









          $endgroup$
















            4












            4








            4





            $begingroup$


            So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.




            Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$



            Conclusion




            All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.







            share|cite|improve this answer









            $endgroup$




            So, we are looking for all $n>5$ for which $5^nequiv 3125=5^5mod 10000$.




            Note that the following equivalence holds for $n>5$:$$5^nequiv 5^5mod 10000\iff \5^{n-4}equiv 5mod 16\iff\5^{n-5}equiv 1mod 16$$Define $mtriangleq n-5ge 1$. Then all the $m$s for which $5^{m}equiv 1mod 16$ holds are $$m=4kquad,quad kin Bbb N$$this is because $5^4=625equiv 1mod 16$ and therefore $$5^{4k}equiv5^{4k-4}equivcdots equiv 5^{4}equiv 1mod 16$$



            Conclusion




            All $n>5$s for which $5^n$ ends up with $3125$ can be found from $$n=4k+5quad,quad kin Bbb N$$ and the smallest such $n$ is 9.








            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 2 hours ago









            Mostafa AyazMostafa Ayaz

            17k3939




            17k3939























                2












                $begingroup$

                Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?






                share|cite|improve this answer









                $endgroup$


















                  2












                  $begingroup$

                  Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?






                  share|cite|improve this answer









                  $endgroup$
















                    2












                    2








                    2





                    $begingroup$

                    Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?






                    share|cite|improve this answer









                    $endgroup$



                    Hint: $5^n equiv 5^5 mod 10^4$ if and only if $5^n equiv 5^5 mod 2^4$. What is the multiplicative order of $5$ mod $16$?







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 hours ago









                    Robert IsraelRobert Israel

                    328k23216469




                    328k23216469























                        0












                        $begingroup$

                        Well
                        $$5^9=1953125$$
                        so the answer is $9$. In fact
                        $$5^nequiv 5^{n-4} mod{10^4}$$
                        For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.






                        share|cite|improve this answer











                        $endgroup$













                        • $begingroup$
                          Why not $5^5 = 3125$.
                          $endgroup$
                          – fleablood
                          2 hours ago






                        • 1




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          2 hours ago
















                        0












                        $begingroup$

                        Well
                        $$5^9=1953125$$
                        so the answer is $9$. In fact
                        $$5^nequiv 5^{n-4} mod{10^4}$$
                        For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.






                        share|cite|improve this answer











                        $endgroup$













                        • $begingroup$
                          Why not $5^5 = 3125$.
                          $endgroup$
                          – fleablood
                          2 hours ago






                        • 1




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          2 hours ago














                        0












                        0








                        0





                        $begingroup$

                        Well
                        $$5^9=1953125$$
                        so the answer is $9$. In fact
                        $$5^nequiv 5^{n-4} mod{10^4}$$
                        For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.






                        share|cite|improve this answer











                        $endgroup$



                        Well
                        $$5^9=1953125$$
                        so the answer is $9$. In fact
                        $$5^nequiv 5^{n-4} mod{10^4}$$
                        For $nge 8$, so any value of $5^{5+4k}$ where $kinmathbb{N}$ has the last four digits $3125$.







                        share|cite|improve this answer














                        share|cite|improve this answer



                        share|cite|improve this answer








                        edited 2 hours ago

























                        answered 2 hours ago









                        Peter ForemanPeter Foreman

                        3,8371216




                        3,8371216












                        • $begingroup$
                          Why not $5^5 = 3125$.
                          $endgroup$
                          – fleablood
                          2 hours ago






                        • 1




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          2 hours ago


















                        • $begingroup$
                          Why not $5^5 = 3125$.
                          $endgroup$
                          – fleablood
                          2 hours ago






                        • 1




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          2 hours ago
















                        $begingroup$
                        Why not $5^5 = 3125$.
                        $endgroup$
                        – fleablood
                        2 hours ago




                        $begingroup$
                        Why not $5^5 = 3125$.
                        $endgroup$
                        – fleablood
                        2 hours ago




                        1




                        1




                        $begingroup$
                        The question states that $ngt5$
                        $endgroup$
                        – Peter Foreman
                        2 hours ago




                        $begingroup$
                        The question states that $ngt5$
                        $endgroup$
                        – Peter Foreman
                        2 hours ago











                        0












                        $begingroup$

                        Hint $, 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}).,$ Now recall $, begin{align} 5, &equiv 1!pmod{! color{#c00}4} \ Rightarrow 5^{largecolor{#c00} 4}!&equiv 1^{largecolor{#c00} 4}!!!! pmod{!color{#c00}4^{large 2}}end{align}$






                        share|cite|improve this answer











                        $endgroup$


















                          0












                          $begingroup$

                          Hint $, 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}).,$ Now recall $, begin{align} 5, &equiv 1!pmod{! color{#c00}4} \ Rightarrow 5^{largecolor{#c00} 4}!&equiv 1^{largecolor{#c00} 4}!!!! pmod{!color{#c00}4^{large 2}}end{align}$






                          share|cite|improve this answer











                          $endgroup$
















                            0












                            0








                            0





                            $begingroup$

                            Hint $, 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}).,$ Now recall $, begin{align} 5, &equiv 1!pmod{! color{#c00}4} \ Rightarrow 5^{largecolor{#c00} 4}!&equiv 1^{largecolor{#c00} 4}!!!! pmod{!color{#c00}4^{large 2}}end{align}$






                            share|cite|improve this answer











                            $endgroup$



                            Hint $, 5^{large 5+N}! bmod 10^{large 4} = 5^{large 5}(5^{largecolor{#c00} N}! bmod 2^{large 4}).,$ Now recall $, begin{align} 5, &equiv 1!pmod{! color{#c00}4} \ Rightarrow 5^{largecolor{#c00} 4}!&equiv 1^{largecolor{#c00} 4}!!!! pmod{!color{#c00}4^{large 2}}end{align}$







                            share|cite|improve this answer














                            share|cite|improve this answer



                            share|cite|improve this answer








                            edited 20 mins ago

























                            answered 1 hour ago









                            Bill DubuqueBill Dubuque

                            212k29195654




                            212k29195654






















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