Find last 3 digits of this monster numberWhen can you simplify the modulus? ($10^5^102 text mod 35$)how to find the last non-zero digit of $n$Why is $x^100 = 1 mod 1000$ if $x < 1000$ and $gcd (x,1000) = 1$?Last 3 digits of Marsenne numbersShow that $a^varphi(n)+1equiv a,mod n$.Showing that $a^5$ and $a$ have the same last digit using Euler's Theorem.Find the last ten digits of this exponential tower.Euler theorem, finding last digitsNumber Theory Linear Diophantine EquationsRSA decryption coefficientFinding the last 4 digits of a huge power

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Find last 3 digits of this monster number


When can you simplify the modulus? ($10^5^102 text mod 35$)how to find the last non-zero digit of $n$Why is $x^100 = 1 mod 1000$ if $x < 1000$ and $gcd (x,1000) = 1$?Last 3 digits of Marsenne numbersShow that $a^varphi(n)+1equiv a,mod n$.Showing that $a^5$ and $a$ have the same last digit using Euler's Theorem.Find the last ten digits of this exponential tower.Euler theorem, finding last digitsNumber Theory Linear Diophantine EquationsRSA decryption coefficientFinding the last 4 digits of a huge power













5












$begingroup$


Find the last 3 digits of this number
$$
2032^2031^2030^dots^2^1
$$

So obviously we are looking for $x$ so that
$$
2032^2031^2030^dots^2^1 equiv x quad textmodhspace0.1cm 1000
$$

I also know that usually you use Euler' theorem here, but that only works when the numbers $a$ and $n$ are coprime, but $2032$ and $1000$ are not coprime? I can easily find $varphi(1000)$, that is not a problem. Am I looking for wrong numbers to be coprime here or is there another way instead of Euler' theorem?










share|cite|improve this question









$endgroup$











  • $begingroup$
    The common factor here is $8$ and you should be able show that he power is divisible by $8$, so the residue modulo $1000$ can be determined by looking at the residue modulo $1000/8=125$
    $endgroup$
    – Mark Bennet
    3 hours ago










  • $begingroup$
    @Mark is correct, and in fact we can factor it out in a slick operational way that avoids using CRT by using the mod Distributive Law, as I show in my answer. This usually ends up being simpler than rotely applying CRT = Chinese Remainder when the base shares a common factor with the modulus.
    $endgroup$
    – Bill Dubuque
    49 mins ago
















5












$begingroup$


Find the last 3 digits of this number
$$
2032^2031^2030^dots^2^1
$$

So obviously we are looking for $x$ so that
$$
2032^2031^2030^dots^2^1 equiv x quad textmodhspace0.1cm 1000
$$

I also know that usually you use Euler' theorem here, but that only works when the numbers $a$ and $n$ are coprime, but $2032$ and $1000$ are not coprime? I can easily find $varphi(1000)$, that is not a problem. Am I looking for wrong numbers to be coprime here or is there another way instead of Euler' theorem?










share|cite|improve this question









$endgroup$











  • $begingroup$
    The common factor here is $8$ and you should be able show that he power is divisible by $8$, so the residue modulo $1000$ can be determined by looking at the residue modulo $1000/8=125$
    $endgroup$
    – Mark Bennet
    3 hours ago










  • $begingroup$
    @Mark is correct, and in fact we can factor it out in a slick operational way that avoids using CRT by using the mod Distributive Law, as I show in my answer. This usually ends up being simpler than rotely applying CRT = Chinese Remainder when the base shares a common factor with the modulus.
    $endgroup$
    – Bill Dubuque
    49 mins ago














5












5








5





$begingroup$


Find the last 3 digits of this number
$$
2032^2031^2030^dots^2^1
$$

So obviously we are looking for $x$ so that
$$
2032^2031^2030^dots^2^1 equiv x quad textmodhspace0.1cm 1000
$$

I also know that usually you use Euler' theorem here, but that only works when the numbers $a$ and $n$ are coprime, but $2032$ and $1000$ are not coprime? I can easily find $varphi(1000)$, that is not a problem. Am I looking for wrong numbers to be coprime here or is there another way instead of Euler' theorem?










share|cite|improve this question









$endgroup$




Find the last 3 digits of this number
$$
2032^2031^2030^dots^2^1
$$

So obviously we are looking for $x$ so that
$$
2032^2031^2030^dots^2^1 equiv x quad textmodhspace0.1cm 1000
$$

I also know that usually you use Euler' theorem here, but that only works when the numbers $a$ and $n$ are coprime, but $2032$ and $1000$ are not coprime? I can easily find $varphi(1000)$, that is not a problem. Am I looking for wrong numbers to be coprime here or is there another way instead of Euler' theorem?







number-theory totient-function






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 3 hours ago









Kristin PeterselKristin Petersel

263




263











  • $begingroup$
    The common factor here is $8$ and you should be able show that he power is divisible by $8$, so the residue modulo $1000$ can be determined by looking at the residue modulo $1000/8=125$
    $endgroup$
    – Mark Bennet
    3 hours ago










  • $begingroup$
    @Mark is correct, and in fact we can factor it out in a slick operational way that avoids using CRT by using the mod Distributive Law, as I show in my answer. This usually ends up being simpler than rotely applying CRT = Chinese Remainder when the base shares a common factor with the modulus.
    $endgroup$
    – Bill Dubuque
    49 mins ago

















  • $begingroup$
    The common factor here is $8$ and you should be able show that he power is divisible by $8$, so the residue modulo $1000$ can be determined by looking at the residue modulo $1000/8=125$
    $endgroup$
    – Mark Bennet
    3 hours ago










  • $begingroup$
    @Mark is correct, and in fact we can factor it out in a slick operational way that avoids using CRT by using the mod Distributive Law, as I show in my answer. This usually ends up being simpler than rotely applying CRT = Chinese Remainder when the base shares a common factor with the modulus.
    $endgroup$
    – Bill Dubuque
    49 mins ago
















$begingroup$
The common factor here is $8$ and you should be able show that he power is divisible by $8$, so the residue modulo $1000$ can be determined by looking at the residue modulo $1000/8=125$
$endgroup$
– Mark Bennet
3 hours ago




$begingroup$
The common factor here is $8$ and you should be able show that he power is divisible by $8$, so the residue modulo $1000$ can be determined by looking at the residue modulo $1000/8=125$
$endgroup$
– Mark Bennet
3 hours ago












$begingroup$
@Mark is correct, and in fact we can factor it out in a slick operational way that avoids using CRT by using the mod Distributive Law, as I show in my answer. This usually ends up being simpler than rotely applying CRT = Chinese Remainder when the base shares a common factor with the modulus.
$endgroup$
– Bill Dubuque
49 mins ago





$begingroup$
@Mark is correct, and in fact we can factor it out in a slick operational way that avoids using CRT by using the mod Distributive Law, as I show in my answer. This usually ends up being simpler than rotely applying CRT = Chinese Remainder when the base shares a common factor with the modulus.
$endgroup$
– Bill Dubuque
49 mins ago











4 Answers
4






active

oldest

votes


















2












$begingroup$

It's a lot simpler than it looks. I shall call the number $N$.



You will know the residue modulo $10^3$, thus the last three digits, if you first get the residues modulo $2^3=8$ and modulo $5^3=125$.



$N$ is obviously a multiple of $8$, thus $Nequiv 0bmod 8$. Which leaves $bmod 125$.



The base $2032equiv 32$. When this is raised to a power, the residue of this power depends only on the residue of the exponent $bmod 100$ where $100$ is the Euler totient of $125$. But the exponent on $2032$ has the form



$2031^10k=(2030+1)^10k=(textbinomial expansion)=100m+1$



So $Nequiv 32^1equiv 32bmod 125$. The only multiple of $8$ between $0$ and $999$ satisfying this result is $32$ itself so ... $Nequiv 32bmod 1000$. Meaning the last three digits were there all along, the $colorblue032$ in the base $2032$!






share|cite|improve this answer











$endgroup$








  • 1




    $begingroup$
    (+1) same answer I got, by means similar enough that I won't add another post here.
    $endgroup$
    – robjohn
    2 hours ago











  • $begingroup$
    Worth emphasis is that arguments like this can be presented completely operationally by employing the mod Distributive Law, and this clarifies and simplifies the arithmetic - see my answer.
    $endgroup$
    – Bill Dubuque
    1 hour ago



















1












$begingroup$

$bmod 1000!: 32^large 2031^LARGE 2k!!equiv, 8left[dfraccolor#0a032^large 2031^LARGE 2k8 bmod color#0a0125right]! equiv 8left[dfraccolor#c00328bmod 125right]! equiv 32, $ by



$ ,beginalign !bmod color#0a0125!: color#0a032^large 2031^LARGE 2k!!
&equiv, 2^large 5cdot 2031^LARGE 2k! bmod 100 rm by 100 = phi(125) rm [Euler totient]\
&equiv,2^large 5(color#b6f2031^LARGE color#d4f2k! bmod 20) rm by mod Distributive Law\
&equiv,2^large 5(color#b6f1^LARGE k)equiv, color#c0032 rm by color#b6f2031^large 2!equiv 11^large 2equivcolor#b6f 1!!!pmod!20\
endalign $






share|cite|improve this answer











$endgroup$








  • 1




    $begingroup$
    We used twice: $, abbmod ac, =, a,(bbmod c), =, $ mod Distributive Law $ $
    $endgroup$
    – Bill Dubuque
    1 hour ago



















0












$begingroup$

By the Chinese Remainder Theorem, if you want to find what remainder a given number has when divided by $1000$, you can split that into 2 problems: Find the remainder$mod 8$ and$mod 125$. Obviously



$$z_0:=2032^2031^2030^dots^2^1 equiv 0 pmod 8$$



What remains to be found is $x_0 in [0,124]$ in



$$z_0 equiv x_0 pmod 125.$$



As $z_0$ is now coprime to $125$, you can apply Euler's theorem now. With



$$z_1:=2031^2030^dots^2^1$$



and



$$phi(125)=100$$ the new problem becomes to find $x_1 in [0,99]$ in



$$z_1 equiv x_1 pmod 100$$



and then use



$$32^x_1 equiv x_0 pmod 125$$



to find $x_0$.



So this reduced the original problem$mod 1000$ to a smaller problem$mod 100$.



Applying this reduction procedure a few more times (using the Chinese Remainder Theorem if appropriate), should result in congruences with smaller and smaller module that can in the end be solved (e.g $mod 2$).



Then to solve the original problem you need to back-substitute the calculated $x_i$ to get $x_i-1$, just as outlined for $x_1,x_0$ above.






share|cite|improve this answer









$endgroup$




















    0












    $begingroup$

    Don't be scared. If it turns into a monster and eats you, run away after you throw stones at it. Don't run away before throwing stones just because it looks like a monster.



    $2032^monster$ and $1000$ are relatively prime so we can't use Euler theorem but we can break it down with Chinese remainder theorem.



    $2032^monster = 0 pmod 8$ and so we just need to solve $2032^monster pmod 125$ and for that we can use Euler Theorem.



    $phi(125=5^3) = (5-1)*5^3-1 = 100$.



    So $2032^monster equiv 32^monster % 100$.



    And $monster = 2031^littlemonsterequiv 31^littlemonsterpmod 100$



    $31$ and $100$ are relatively prime and $phi(100)= 40$ so



    $31^littlemonster equiv 31^littlemonster % 40 pmod 100$.



    $littlemonster = 2030^smallmonster$ but $5|2030$ and as $smallmonster > 2$ we know $8|2^smallmonster$ and $2^smallmonster|2030^smallmonster$.



    So $littlemonster equiv 0 pmod 40$.



    $2031^littlemonster equiv 31^0 equiv 1 pmod 100$



    So $2032^monster equiv 31 pmod 125$



    So $2032^monster equiv 0 pmod 8$ and $2032^monster equiv 31 pmod 125$.



    So $2032^monster equiv 31 + 125k pmod 1000$ where $8|31 + 125k$.



    I.e. $31+125k equiv -1 - 3k equiv 0 pmod 8$ so $3k equiv -1equiv 15 pmod 8$ so $k=5$ and



    $2032^monster equiv 656pmod1000$






    share|cite|improve this answer









    $endgroup$












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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      2












      $begingroup$

      It's a lot simpler than it looks. I shall call the number $N$.



      You will know the residue modulo $10^3$, thus the last three digits, if you first get the residues modulo $2^3=8$ and modulo $5^3=125$.



      $N$ is obviously a multiple of $8$, thus $Nequiv 0bmod 8$. Which leaves $bmod 125$.



      The base $2032equiv 32$. When this is raised to a power, the residue of this power depends only on the residue of the exponent $bmod 100$ where $100$ is the Euler totient of $125$. But the exponent on $2032$ has the form



      $2031^10k=(2030+1)^10k=(textbinomial expansion)=100m+1$



      So $Nequiv 32^1equiv 32bmod 125$. The only multiple of $8$ between $0$ and $999$ satisfying this result is $32$ itself so ... $Nequiv 32bmod 1000$. Meaning the last three digits were there all along, the $colorblue032$ in the base $2032$!






      share|cite|improve this answer











      $endgroup$








      • 1




        $begingroup$
        (+1) same answer I got, by means similar enough that I won't add another post here.
        $endgroup$
        – robjohn
        2 hours ago











      • $begingroup$
        Worth emphasis is that arguments like this can be presented completely operationally by employing the mod Distributive Law, and this clarifies and simplifies the arithmetic - see my answer.
        $endgroup$
        – Bill Dubuque
        1 hour ago
















      2












      $begingroup$

      It's a lot simpler than it looks. I shall call the number $N$.



      You will know the residue modulo $10^3$, thus the last three digits, if you first get the residues modulo $2^3=8$ and modulo $5^3=125$.



      $N$ is obviously a multiple of $8$, thus $Nequiv 0bmod 8$. Which leaves $bmod 125$.



      The base $2032equiv 32$. When this is raised to a power, the residue of this power depends only on the residue of the exponent $bmod 100$ where $100$ is the Euler totient of $125$. But the exponent on $2032$ has the form



      $2031^10k=(2030+1)^10k=(textbinomial expansion)=100m+1$



      So $Nequiv 32^1equiv 32bmod 125$. The only multiple of $8$ between $0$ and $999$ satisfying this result is $32$ itself so ... $Nequiv 32bmod 1000$. Meaning the last three digits were there all along, the $colorblue032$ in the base $2032$!






      share|cite|improve this answer











      $endgroup$








      • 1




        $begingroup$
        (+1) same answer I got, by means similar enough that I won't add another post here.
        $endgroup$
        – robjohn
        2 hours ago











      • $begingroup$
        Worth emphasis is that arguments like this can be presented completely operationally by employing the mod Distributive Law, and this clarifies and simplifies the arithmetic - see my answer.
        $endgroup$
        – Bill Dubuque
        1 hour ago














      2












      2








      2





      $begingroup$

      It's a lot simpler than it looks. I shall call the number $N$.



      You will know the residue modulo $10^3$, thus the last three digits, if you first get the residues modulo $2^3=8$ and modulo $5^3=125$.



      $N$ is obviously a multiple of $8$, thus $Nequiv 0bmod 8$. Which leaves $bmod 125$.



      The base $2032equiv 32$. When this is raised to a power, the residue of this power depends only on the residue of the exponent $bmod 100$ where $100$ is the Euler totient of $125$. But the exponent on $2032$ has the form



      $2031^10k=(2030+1)^10k=(textbinomial expansion)=100m+1$



      So $Nequiv 32^1equiv 32bmod 125$. The only multiple of $8$ between $0$ and $999$ satisfying this result is $32$ itself so ... $Nequiv 32bmod 1000$. Meaning the last three digits were there all along, the $colorblue032$ in the base $2032$!






      share|cite|improve this answer











      $endgroup$



      It's a lot simpler than it looks. I shall call the number $N$.



      You will know the residue modulo $10^3$, thus the last three digits, if you first get the residues modulo $2^3=8$ and modulo $5^3=125$.



      $N$ is obviously a multiple of $8$, thus $Nequiv 0bmod 8$. Which leaves $bmod 125$.



      The base $2032equiv 32$. When this is raised to a power, the residue of this power depends only on the residue of the exponent $bmod 100$ where $100$ is the Euler totient of $125$. But the exponent on $2032$ has the form



      $2031^10k=(2030+1)^10k=(textbinomial expansion)=100m+1$



      So $Nequiv 32^1equiv 32bmod 125$. The only multiple of $8$ between $0$ and $999$ satisfying this result is $32$ itself so ... $Nequiv 32bmod 1000$. Meaning the last three digits were there all along, the $colorblue032$ in the base $2032$!







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited 3 hours ago

























      answered 3 hours ago









      Oscar LanziOscar Lanzi

      13.3k12136




      13.3k12136







      • 1




        $begingroup$
        (+1) same answer I got, by means similar enough that I won't add another post here.
        $endgroup$
        – robjohn
        2 hours ago











      • $begingroup$
        Worth emphasis is that arguments like this can be presented completely operationally by employing the mod Distributive Law, and this clarifies and simplifies the arithmetic - see my answer.
        $endgroup$
        – Bill Dubuque
        1 hour ago













      • 1




        $begingroup$
        (+1) same answer I got, by means similar enough that I won't add another post here.
        $endgroup$
        – robjohn
        2 hours ago











      • $begingroup$
        Worth emphasis is that arguments like this can be presented completely operationally by employing the mod Distributive Law, and this clarifies and simplifies the arithmetic - see my answer.
        $endgroup$
        – Bill Dubuque
        1 hour ago








      1




      1




      $begingroup$
      (+1) same answer I got, by means similar enough that I won't add another post here.
      $endgroup$
      – robjohn
      2 hours ago





      $begingroup$
      (+1) same answer I got, by means similar enough that I won't add another post here.
      $endgroup$
      – robjohn
      2 hours ago













      $begingroup$
      Worth emphasis is that arguments like this can be presented completely operationally by employing the mod Distributive Law, and this clarifies and simplifies the arithmetic - see my answer.
      $endgroup$
      – Bill Dubuque
      1 hour ago





      $begingroup$
      Worth emphasis is that arguments like this can be presented completely operationally by employing the mod Distributive Law, and this clarifies and simplifies the arithmetic - see my answer.
      $endgroup$
      – Bill Dubuque
      1 hour ago












      1












      $begingroup$

      $bmod 1000!: 32^large 2031^LARGE 2k!!equiv, 8left[dfraccolor#0a032^large 2031^LARGE 2k8 bmod color#0a0125right]! equiv 8left[dfraccolor#c00328bmod 125right]! equiv 32, $ by



      $ ,beginalign !bmod color#0a0125!: color#0a032^large 2031^LARGE 2k!!
      &equiv, 2^large 5cdot 2031^LARGE 2k! bmod 100 rm by 100 = phi(125) rm [Euler totient]\
      &equiv,2^large 5(color#b6f2031^LARGE color#d4f2k! bmod 20) rm by mod Distributive Law\
      &equiv,2^large 5(color#b6f1^LARGE k)equiv, color#c0032 rm by color#b6f2031^large 2!equiv 11^large 2equivcolor#b6f 1!!!pmod!20\
      endalign $






      share|cite|improve this answer











      $endgroup$








      • 1




        $begingroup$
        We used twice: $, abbmod ac, =, a,(bbmod c), =, $ mod Distributive Law $ $
        $endgroup$
        – Bill Dubuque
        1 hour ago
















      1












      $begingroup$

      $bmod 1000!: 32^large 2031^LARGE 2k!!equiv, 8left[dfraccolor#0a032^large 2031^LARGE 2k8 bmod color#0a0125right]! equiv 8left[dfraccolor#c00328bmod 125right]! equiv 32, $ by



      $ ,beginalign !bmod color#0a0125!: color#0a032^large 2031^LARGE 2k!!
      &equiv, 2^large 5cdot 2031^LARGE 2k! bmod 100 rm by 100 = phi(125) rm [Euler totient]\
      &equiv,2^large 5(color#b6f2031^LARGE color#d4f2k! bmod 20) rm by mod Distributive Law\
      &equiv,2^large 5(color#b6f1^LARGE k)equiv, color#c0032 rm by color#b6f2031^large 2!equiv 11^large 2equivcolor#b6f 1!!!pmod!20\
      endalign $






      share|cite|improve this answer











      $endgroup$








      • 1




        $begingroup$
        We used twice: $, abbmod ac, =, a,(bbmod c), =, $ mod Distributive Law $ $
        $endgroup$
        – Bill Dubuque
        1 hour ago














      1












      1








      1





      $begingroup$

      $bmod 1000!: 32^large 2031^LARGE 2k!!equiv, 8left[dfraccolor#0a032^large 2031^LARGE 2k8 bmod color#0a0125right]! equiv 8left[dfraccolor#c00328bmod 125right]! equiv 32, $ by



      $ ,beginalign !bmod color#0a0125!: color#0a032^large 2031^LARGE 2k!!
      &equiv, 2^large 5cdot 2031^LARGE 2k! bmod 100 rm by 100 = phi(125) rm [Euler totient]\
      &equiv,2^large 5(color#b6f2031^LARGE color#d4f2k! bmod 20) rm by mod Distributive Law\
      &equiv,2^large 5(color#b6f1^LARGE k)equiv, color#c0032 rm by color#b6f2031^large 2!equiv 11^large 2equivcolor#b6f 1!!!pmod!20\
      endalign $






      share|cite|improve this answer











      $endgroup$



      $bmod 1000!: 32^large 2031^LARGE 2k!!equiv, 8left[dfraccolor#0a032^large 2031^LARGE 2k8 bmod color#0a0125right]! equiv 8left[dfraccolor#c00328bmod 125right]! equiv 32, $ by



      $ ,beginalign !bmod color#0a0125!: color#0a032^large 2031^LARGE 2k!!
      &equiv, 2^large 5cdot 2031^LARGE 2k! bmod 100 rm by 100 = phi(125) rm [Euler totient]\
      &equiv,2^large 5(color#b6f2031^LARGE color#d4f2k! bmod 20) rm by mod Distributive Law\
      &equiv,2^large 5(color#b6f1^LARGE k)equiv, color#c0032 rm by color#b6f2031^large 2!equiv 11^large 2equivcolor#b6f 1!!!pmod!20\
      endalign $







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited 1 hour ago

























      answered 2 hours ago









      Bill DubuqueBill Dubuque

      213k29195654




      213k29195654







      • 1




        $begingroup$
        We used twice: $, abbmod ac, =, a,(bbmod c), =, $ mod Distributive Law $ $
        $endgroup$
        – Bill Dubuque
        1 hour ago













      • 1




        $begingroup$
        We used twice: $, abbmod ac, =, a,(bbmod c), =, $ mod Distributive Law $ $
        $endgroup$
        – Bill Dubuque
        1 hour ago








      1




      1




      $begingroup$
      We used twice: $, abbmod ac, =, a,(bbmod c), =, $ mod Distributive Law $ $
      $endgroup$
      – Bill Dubuque
      1 hour ago





      $begingroup$
      We used twice: $, abbmod ac, =, a,(bbmod c), =, $ mod Distributive Law $ $
      $endgroup$
      – Bill Dubuque
      1 hour ago












      0












      $begingroup$

      By the Chinese Remainder Theorem, if you want to find what remainder a given number has when divided by $1000$, you can split that into 2 problems: Find the remainder$mod 8$ and$mod 125$. Obviously



      $$z_0:=2032^2031^2030^dots^2^1 equiv 0 pmod 8$$



      What remains to be found is $x_0 in [0,124]$ in



      $$z_0 equiv x_0 pmod 125.$$



      As $z_0$ is now coprime to $125$, you can apply Euler's theorem now. With



      $$z_1:=2031^2030^dots^2^1$$



      and



      $$phi(125)=100$$ the new problem becomes to find $x_1 in [0,99]$ in



      $$z_1 equiv x_1 pmod 100$$



      and then use



      $$32^x_1 equiv x_0 pmod 125$$



      to find $x_0$.



      So this reduced the original problem$mod 1000$ to a smaller problem$mod 100$.



      Applying this reduction procedure a few more times (using the Chinese Remainder Theorem if appropriate), should result in congruences with smaller and smaller module that can in the end be solved (e.g $mod 2$).



      Then to solve the original problem you need to back-substitute the calculated $x_i$ to get $x_i-1$, just as outlined for $x_1,x_0$ above.






      share|cite|improve this answer









      $endgroup$

















        0












        $begingroup$

        By the Chinese Remainder Theorem, if you want to find what remainder a given number has when divided by $1000$, you can split that into 2 problems: Find the remainder$mod 8$ and$mod 125$. Obviously



        $$z_0:=2032^2031^2030^dots^2^1 equiv 0 pmod 8$$



        What remains to be found is $x_0 in [0,124]$ in



        $$z_0 equiv x_0 pmod 125.$$



        As $z_0$ is now coprime to $125$, you can apply Euler's theorem now. With



        $$z_1:=2031^2030^dots^2^1$$



        and



        $$phi(125)=100$$ the new problem becomes to find $x_1 in [0,99]$ in



        $$z_1 equiv x_1 pmod 100$$



        and then use



        $$32^x_1 equiv x_0 pmod 125$$



        to find $x_0$.



        So this reduced the original problem$mod 1000$ to a smaller problem$mod 100$.



        Applying this reduction procedure a few more times (using the Chinese Remainder Theorem if appropriate), should result in congruences with smaller and smaller module that can in the end be solved (e.g $mod 2$).



        Then to solve the original problem you need to back-substitute the calculated $x_i$ to get $x_i-1$, just as outlined for $x_1,x_0$ above.






        share|cite|improve this answer









        $endgroup$















          0












          0








          0





          $begingroup$

          By the Chinese Remainder Theorem, if you want to find what remainder a given number has when divided by $1000$, you can split that into 2 problems: Find the remainder$mod 8$ and$mod 125$. Obviously



          $$z_0:=2032^2031^2030^dots^2^1 equiv 0 pmod 8$$



          What remains to be found is $x_0 in [0,124]$ in



          $$z_0 equiv x_0 pmod 125.$$



          As $z_0$ is now coprime to $125$, you can apply Euler's theorem now. With



          $$z_1:=2031^2030^dots^2^1$$



          and



          $$phi(125)=100$$ the new problem becomes to find $x_1 in [0,99]$ in



          $$z_1 equiv x_1 pmod 100$$



          and then use



          $$32^x_1 equiv x_0 pmod 125$$



          to find $x_0$.



          So this reduced the original problem$mod 1000$ to a smaller problem$mod 100$.



          Applying this reduction procedure a few more times (using the Chinese Remainder Theorem if appropriate), should result in congruences with smaller and smaller module that can in the end be solved (e.g $mod 2$).



          Then to solve the original problem you need to back-substitute the calculated $x_i$ to get $x_i-1$, just as outlined for $x_1,x_0$ above.






          share|cite|improve this answer









          $endgroup$



          By the Chinese Remainder Theorem, if you want to find what remainder a given number has when divided by $1000$, you can split that into 2 problems: Find the remainder$mod 8$ and$mod 125$. Obviously



          $$z_0:=2032^2031^2030^dots^2^1 equiv 0 pmod 8$$



          What remains to be found is $x_0 in [0,124]$ in



          $$z_0 equiv x_0 pmod 125.$$



          As $z_0$ is now coprime to $125$, you can apply Euler's theorem now. With



          $$z_1:=2031^2030^dots^2^1$$



          and



          $$phi(125)=100$$ the new problem becomes to find $x_1 in [0,99]$ in



          $$z_1 equiv x_1 pmod 100$$



          and then use



          $$32^x_1 equiv x_0 pmod 125$$



          to find $x_0$.



          So this reduced the original problem$mod 1000$ to a smaller problem$mod 100$.



          Applying this reduction procedure a few more times (using the Chinese Remainder Theorem if appropriate), should result in congruences with smaller and smaller module that can in the end be solved (e.g $mod 2$).



          Then to solve the original problem you need to back-substitute the calculated $x_i$ to get $x_i-1$, just as outlined for $x_1,x_0$ above.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 3 hours ago









          IngixIngix

          5,032159




          5,032159





















              0












              $begingroup$

              Don't be scared. If it turns into a monster and eats you, run away after you throw stones at it. Don't run away before throwing stones just because it looks like a monster.



              $2032^monster$ and $1000$ are relatively prime so we can't use Euler theorem but we can break it down with Chinese remainder theorem.



              $2032^monster = 0 pmod 8$ and so we just need to solve $2032^monster pmod 125$ and for that we can use Euler Theorem.



              $phi(125=5^3) = (5-1)*5^3-1 = 100$.



              So $2032^monster equiv 32^monster % 100$.



              And $monster = 2031^littlemonsterequiv 31^littlemonsterpmod 100$



              $31$ and $100$ are relatively prime and $phi(100)= 40$ so



              $31^littlemonster equiv 31^littlemonster % 40 pmod 100$.



              $littlemonster = 2030^smallmonster$ but $5|2030$ and as $smallmonster > 2$ we know $8|2^smallmonster$ and $2^smallmonster|2030^smallmonster$.



              So $littlemonster equiv 0 pmod 40$.



              $2031^littlemonster equiv 31^0 equiv 1 pmod 100$



              So $2032^monster equiv 31 pmod 125$



              So $2032^monster equiv 0 pmod 8$ and $2032^monster equiv 31 pmod 125$.



              So $2032^monster equiv 31 + 125k pmod 1000$ where $8|31 + 125k$.



              I.e. $31+125k equiv -1 - 3k equiv 0 pmod 8$ so $3k equiv -1equiv 15 pmod 8$ so $k=5$ and



              $2032^monster equiv 656pmod1000$






              share|cite|improve this answer









              $endgroup$

















                0












                $begingroup$

                Don't be scared. If it turns into a monster and eats you, run away after you throw stones at it. Don't run away before throwing stones just because it looks like a monster.



                $2032^monster$ and $1000$ are relatively prime so we can't use Euler theorem but we can break it down with Chinese remainder theorem.



                $2032^monster = 0 pmod 8$ and so we just need to solve $2032^monster pmod 125$ and for that we can use Euler Theorem.



                $phi(125=5^3) = (5-1)*5^3-1 = 100$.



                So $2032^monster equiv 32^monster % 100$.



                And $monster = 2031^littlemonsterequiv 31^littlemonsterpmod 100$



                $31$ and $100$ are relatively prime and $phi(100)= 40$ so



                $31^littlemonster equiv 31^littlemonster % 40 pmod 100$.



                $littlemonster = 2030^smallmonster$ but $5|2030$ and as $smallmonster > 2$ we know $8|2^smallmonster$ and $2^smallmonster|2030^smallmonster$.



                So $littlemonster equiv 0 pmod 40$.



                $2031^littlemonster equiv 31^0 equiv 1 pmod 100$



                So $2032^monster equiv 31 pmod 125$



                So $2032^monster equiv 0 pmod 8$ and $2032^monster equiv 31 pmod 125$.



                So $2032^monster equiv 31 + 125k pmod 1000$ where $8|31 + 125k$.



                I.e. $31+125k equiv -1 - 3k equiv 0 pmod 8$ so $3k equiv -1equiv 15 pmod 8$ so $k=5$ and



                $2032^monster equiv 656pmod1000$






                share|cite|improve this answer









                $endgroup$















                  0












                  0








                  0





                  $begingroup$

                  Don't be scared. If it turns into a monster and eats you, run away after you throw stones at it. Don't run away before throwing stones just because it looks like a monster.



                  $2032^monster$ and $1000$ are relatively prime so we can't use Euler theorem but we can break it down with Chinese remainder theorem.



                  $2032^monster = 0 pmod 8$ and so we just need to solve $2032^monster pmod 125$ and for that we can use Euler Theorem.



                  $phi(125=5^3) = (5-1)*5^3-1 = 100$.



                  So $2032^monster equiv 32^monster % 100$.



                  And $monster = 2031^littlemonsterequiv 31^littlemonsterpmod 100$



                  $31$ and $100$ are relatively prime and $phi(100)= 40$ so



                  $31^littlemonster equiv 31^littlemonster % 40 pmod 100$.



                  $littlemonster = 2030^smallmonster$ but $5|2030$ and as $smallmonster > 2$ we know $8|2^smallmonster$ and $2^smallmonster|2030^smallmonster$.



                  So $littlemonster equiv 0 pmod 40$.



                  $2031^littlemonster equiv 31^0 equiv 1 pmod 100$



                  So $2032^monster equiv 31 pmod 125$



                  So $2032^monster equiv 0 pmod 8$ and $2032^monster equiv 31 pmod 125$.



                  So $2032^monster equiv 31 + 125k pmod 1000$ where $8|31 + 125k$.



                  I.e. $31+125k equiv -1 - 3k equiv 0 pmod 8$ so $3k equiv -1equiv 15 pmod 8$ so $k=5$ and



                  $2032^monster equiv 656pmod1000$






                  share|cite|improve this answer









                  $endgroup$



                  Don't be scared. If it turns into a monster and eats you, run away after you throw stones at it. Don't run away before throwing stones just because it looks like a monster.



                  $2032^monster$ and $1000$ are relatively prime so we can't use Euler theorem but we can break it down with Chinese remainder theorem.



                  $2032^monster = 0 pmod 8$ and so we just need to solve $2032^monster pmod 125$ and for that we can use Euler Theorem.



                  $phi(125=5^3) = (5-1)*5^3-1 = 100$.



                  So $2032^monster equiv 32^monster % 100$.



                  And $monster = 2031^littlemonsterequiv 31^littlemonsterpmod 100$



                  $31$ and $100$ are relatively prime and $phi(100)= 40$ so



                  $31^littlemonster equiv 31^littlemonster % 40 pmod 100$.



                  $littlemonster = 2030^smallmonster$ but $5|2030$ and as $smallmonster > 2$ we know $8|2^smallmonster$ and $2^smallmonster|2030^smallmonster$.



                  So $littlemonster equiv 0 pmod 40$.



                  $2031^littlemonster equiv 31^0 equiv 1 pmod 100$



                  So $2032^monster equiv 31 pmod 125$



                  So $2032^monster equiv 0 pmod 8$ and $2032^monster equiv 31 pmod 125$.



                  So $2032^monster equiv 31 + 125k pmod 1000$ where $8|31 + 125k$.



                  I.e. $31+125k equiv -1 - 3k equiv 0 pmod 8$ so $3k equiv -1equiv 15 pmod 8$ so $k=5$ and



                  $2032^monster equiv 656pmod1000$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 1 hour ago









                  fleabloodfleablood

                  73.1k22789




                  73.1k22789



























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