Find $438^87493 equiv ~? pmod11$ [closed]How do I compute $a^b,bmod c$ by hand?How to divide $2x equiv 4 pmod 7$ to get just $x equiv Box pmod7$Solving $x^5 equiv 7 mod 13$Infinite exponentiation $n^n^n^…^n equiv m pmod q$ , find m?Modular arithmetic system $x equiv 2 pmod7$ and $x equiv -5 pmod22$How do I find the smallest positive integer $a$ for which $a^n equiv x pmod2^w$?Modular arithmetic problem: $7^x equiv 1 pmod26$Find $6^273 + 8^273pmod49$If $a^2 equiv 4 pmod5$, then $a equiv 2 pmod5$.Solve $x^2 equiv 1 pmod30$ and generalize.Solving $m^3 equiv n^6 pmod19$

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Find $438^87493 equiv ~? pmod11$ [closed]


How do I compute $a^b,bmod c$ by hand?How to divide $2x equiv 4 pmod 7$ to get just $x equiv Box pmod7$Solving $x^5 equiv 7 mod 13$Infinite exponentiation $n^n^n^…^n equiv m pmod q$ , find m?Modular arithmetic system $x equiv 2 pmod7$ and $x equiv -5 pmod22$How do I find the smallest positive integer $a$ for which $a^n equiv x pmod2^w$?Modular arithmetic problem: $7^x equiv 1 pmod26$Find $6^273 + 8^273pmod49$If $a^2 equiv 4 pmod5$, then $a equiv 2 pmod5$.Solve $x^2 equiv 1 pmod30$ and generalize.Solving $m^3 equiv n^6 pmod19$













-1












$begingroup$


How to find the value of '?'

a mod m = b mod m , will this formula be used?

I am taking discrete maths course for CS. And this question is from one of its chapter










share|cite|improve this question











$endgroup$



closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.











  • 2




    $begingroup$
    What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:39











  • $begingroup$
    438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
    $endgroup$
    – rdr2
    Mar 21 at 6:43










  • $begingroup$
    For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:44










  • $begingroup$
    I am taking discrete maths course for CS. And this question is from one of its chapter
    $endgroup$
    – rdr2
    Mar 21 at 6:47







  • 1




    $begingroup$
    Possible duplicate of How do I compute $a^b,bmod c$ by hand?
    $endgroup$
    – Arnaud D.
    Mar 21 at 10:47















-1












$begingroup$


How to find the value of '?'

a mod m = b mod m , will this formula be used?

I am taking discrete maths course for CS. And this question is from one of its chapter










share|cite|improve this question











$endgroup$



closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.











  • 2




    $begingroup$
    What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:39











  • $begingroup$
    438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
    $endgroup$
    – rdr2
    Mar 21 at 6:43










  • $begingroup$
    For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:44










  • $begingroup$
    I am taking discrete maths course for CS. And this question is from one of its chapter
    $endgroup$
    – rdr2
    Mar 21 at 6:47







  • 1




    $begingroup$
    Possible duplicate of How do I compute $a^b,bmod c$ by hand?
    $endgroup$
    – Arnaud D.
    Mar 21 at 10:47













-1












-1








-1


1



$begingroup$


How to find the value of '?'

a mod m = b mod m , will this formula be used?

I am taking discrete maths course for CS. And this question is from one of its chapter










share|cite|improve this question











$endgroup$




How to find the value of '?'

a mod m = b mod m , will this formula be used?

I am taking discrete maths course for CS. And this question is from one of its chapter







modular-arithmetic






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 21 at 6:51







rdr2

















asked Mar 21 at 6:32









rdr2rdr2

12




12




closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.







closed as off-topic by José Carlos Santos, Saad, Cesareo, Arnaud D., Javi Mar 21 at 12:46


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Saad, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.







  • 2




    $begingroup$
    What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:39











  • $begingroup$
    438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
    $endgroup$
    – rdr2
    Mar 21 at 6:43










  • $begingroup$
    For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:44










  • $begingroup$
    I am taking discrete maths course for CS. And this question is from one of its chapter
    $endgroup$
    – rdr2
    Mar 21 at 6:47







  • 1




    $begingroup$
    Possible duplicate of How do I compute $a^b,bmod c$ by hand?
    $endgroup$
    – Arnaud D.
    Mar 21 at 10:47












  • 2




    $begingroup$
    What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:39











  • $begingroup$
    438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
    $endgroup$
    – rdr2
    Mar 21 at 6:43










  • $begingroup$
    For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
    $endgroup$
    – Robert Shore
    Mar 21 at 6:44










  • $begingroup$
    I am taking discrete maths course for CS. And this question is from one of its chapter
    $endgroup$
    – rdr2
    Mar 21 at 6:47







  • 1




    $begingroup$
    Possible duplicate of How do I compute $a^b,bmod c$ by hand?
    $endgroup$
    – Arnaud D.
    Mar 21 at 10:47







2




2




$begingroup$
What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
$endgroup$
– Robert Shore
Mar 21 at 6:39





$begingroup$
What is $438 (mod 11)$? And what is the order of $(Bbb Z / 11 Bbb Z)^*$?
$endgroup$
– Robert Shore
Mar 21 at 6:39













$begingroup$
438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
$endgroup$
– rdr2
Mar 21 at 6:43




$begingroup$
438 mod 11 would be 9 And I don't know what you mean by order of (Z/11Z)
$endgroup$
– rdr2
Mar 21 at 6:43












$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
$endgroup$
– Robert Shore
Mar 21 at 6:44




$begingroup$
For us to be able to help you effectively, you need to provide more background. Why is this problem of interest to you? How have you tried to solve it?
$endgroup$
– Robert Shore
Mar 21 at 6:44












$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47





$begingroup$
I am taking discrete maths course for CS. And this question is from one of its chapter
$endgroup$
– rdr2
Mar 21 at 6:47





1




1




$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
$endgroup$
– Arnaud D.
Mar 21 at 10:47




$begingroup$
Possible duplicate of How do I compute $a^b,bmod c$ by hand?
$endgroup$
– Arnaud D.
Mar 21 at 10:47










1 Answer
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$begingroup$

Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$



If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$



Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get



$(1)$ $438^87480 equiv 1 (text mod 11).$



$(2)$ $438^13 equiv 3 (text mod 11).$



Therefore what is $438^87493 equiv ~? (text mod 11)$?





$$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$








share|cite|improve this answer











$endgroup$



















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$



    If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$



    Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get



    $(1)$ $438^87480 equiv 1 (text mod 11).$



    $(2)$ $438^13 equiv 3 (text mod 11).$



    Therefore what is $438^87493 equiv ~? (text mod 11)$?





    $$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$








    share|cite|improve this answer











    $endgroup$

















      2












      $begingroup$

      Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$



      If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$



      Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get



      $(1)$ $438^87480 equiv 1 (text mod 11).$



      $(2)$ $438^13 equiv 3 (text mod 11).$



      Therefore what is $438^87493 equiv ~? (text mod 11)$?





      $$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$








      share|cite|improve this answer











      $endgroup$















        2












        2








        2





        $begingroup$

        Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$



        If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$



        Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get



        $(1)$ $438^87480 equiv 1 (text mod 11).$



        $(2)$ $438^13 equiv 3 (text mod 11).$



        Therefore what is $438^87493 equiv ~? (text mod 11)$?





        $$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$








        share|cite|improve this answer











        $endgroup$



        Observe that $text gcd (11,438) = 1.$ So by Euler's theorem $438^10 equiv 1 (text mod 11).$ So $438^87490 equiv 1 (text mod 11).$ Also $438 equiv -2 (text mod 11) implies 438^3 equiv -8 equiv 3 (text mod 11).$ Therefore $$438^87493 equiv 3 (text mod 11).$$



        If you don't know Euler's theorem or Fermat's little theorem then observe from my above calculation that $438^3 equiv 3 (text mod 11).$ So $438^6 equiv 9 equiv -2 (text mod 11) implies 438^30 equiv (-2)^5 equiv -32 equiv 1 (text mod 11).$



        Observe that $87493 = 87480 + 13.$ Since $30 mid 87480$ so $438^87480 equiv 1 (text mod 11).$ Since $438^6 equiv -2 (text mod 11)$ so $438^12 equiv 4 (text mod 11).$ Again $438 equiv -2 (text mod 11).$ So $438^13 equiv 438^12 cdot 438 equiv 4 cdot (-2) equiv -8 equiv 3 (text mod 11).$ Thus we get



        $(1)$ $438^87480 equiv 1 (text mod 11).$



        $(2)$ $438^13 equiv 3 (text mod 11).$



        Therefore what is $438^87493 equiv ~? (text mod 11)$?





        $$438^87493 equiv 438^87480+13 equiv 438^87490 cdot 438^13 equiv 1 cdot 3 equiv 3 (text mod 11).$$









        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Mar 21 at 8:03

























        answered Mar 21 at 6:45









        Dbchatto67Dbchatto67

        2,445522




        2,445522













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