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












2












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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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Catherine Cooper is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • 1




    $begingroup$
    What is your take on this?
    $endgroup$
    – ADITYA PRAKASH
    Mar 20 at 20:01






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    Mar 20 at 20:02






  • 4




    $begingroup$
    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
    Mar 20 at 20:20










  • $begingroup$
    The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
    $endgroup$
    – smci
    Mar 20 at 23:55


















2












$begingroup$


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
    Mar 20 at 20:01






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    Mar 20 at 20:02






  • 4




    $begingroup$
    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
    Mar 20 at 20:20










  • $begingroup$
    The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
    $endgroup$
    – smci
    Mar 20 at 23:55
















2












2








2


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 Mar 20 at 19:59









Catherine Cooper Catherine Cooper

243




243




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
    Mar 20 at 20:01






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    Mar 20 at 20:02






  • 4




    $begingroup$
    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
    Mar 20 at 20:20










  • $begingroup$
    The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
    $endgroup$
    – smci
    Mar 20 at 23:55
















  • 1




    $begingroup$
    What is your take on this?
    $endgroup$
    – ADITYA PRAKASH
    Mar 20 at 20:01






  • 1




    $begingroup$
    Why not just list them out and find it?
    $endgroup$
    – Jair Taylor
    Mar 20 at 20:02






  • 4




    $begingroup$
    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
    Mar 20 at 20:20










  • $begingroup$
    The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
    $endgroup$
    – smci
    Mar 20 at 23:55










1




1




$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
Mar 20 at 20:01




$begingroup$
What is your take on this?
$endgroup$
– ADITYA PRAKASH
Mar 20 at 20:01




1




1




$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
Mar 20 at 20:02




$begingroup$
Why not just list them out and find it?
$endgroup$
– Jair Taylor
Mar 20 at 20:02




4




4




$begingroup$
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
Mar 20 at 20:20




$begingroup$
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
Mar 20 at 20:20












$begingroup$
The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
$endgroup$
– smci
Mar 20 at 23:55






$begingroup$
The answer to "What is the smallest such n>5?" is easy, so you might as well retitle the question "What are all n>5 such that...?"
$endgroup$
– smci
Mar 20 at 23:55












4 Answers
4






active

oldest

votes


















5












$begingroup$


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









$endgroup$





















    3












    $begingroup$

    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









    $endgroup$





















      1












      $begingroup$

      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
        Mar 20 at 20:20






      • 2




        $begingroup$
        The question states that $ngt5$
        $endgroup$
        – Peter Foreman
        Mar 20 at 20:22



















      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











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        4 Answers
        4






        active

        oldest

        votes








        4 Answers
        4






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        5












        $begingroup$


        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









        $endgroup$


















          5












          $begingroup$


          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









          $endgroup$
















            5












            5








            5





            $begingroup$


            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 Mar 20 at 20:17









            Mostafa AyazMostafa Ayaz

            17.8k31039




            17.8k31039























                3












                $begingroup$

                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









                $endgroup$


















                  3












                  $begingroup$

                  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









                  $endgroup$
















                    3












                    3








                    3





                    $begingroup$

                    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 Mar 20 at 20:12









                    Robert IsraelRobert Israel

                    329k23217470




                    329k23217470























                        1












                        $begingroup$

                        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
                          Mar 20 at 20:20






                        • 2




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          Mar 20 at 20:22
















                        1












                        $begingroup$

                        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
                          Mar 20 at 20:20






                        • 2




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          Mar 20 at 20:22














                        1












                        1








                        1





                        $begingroup$

                        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 Mar 20 at 20:16

























                        answered Mar 20 at 20:11









                        Peter ForemanPeter Foreman

                        4,2421216




                        4,2421216












                        • $begingroup$
                          Why not $5^5 = 3125$.
                          $endgroup$
                          – fleablood
                          Mar 20 at 20:20






                        • 2




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          Mar 20 at 20:22


















                        • $begingroup$
                          Why not $5^5 = 3125$.
                          $endgroup$
                          – fleablood
                          Mar 20 at 20:20






                        • 2




                          $begingroup$
                          The question states that $ngt5$
                          $endgroup$
                          – Peter Foreman
                          Mar 20 at 20:22
















                        $begingroup$
                        Why not $5^5 = 3125$.
                        $endgroup$
                        – fleablood
                        Mar 20 at 20:20




                        $begingroup$
                        Why not $5^5 = 3125$.
                        $endgroup$
                        – fleablood
                        Mar 20 at 20:20




                        2




                        2




                        $begingroup$
                        The question states that $ngt5$
                        $endgroup$
                        – Peter Foreman
                        Mar 20 at 20:22




                        $begingroup$
                        The question states that $ngt5$
                        $endgroup$
                        – Peter Foreman
                        Mar 20 at 20:22











                        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 Mar 20 at 22:24

























                            answered Mar 20 at 21:34









                            Bill DubuqueBill Dubuque

                            213k29195654




                            213k29195654






















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