Find last 3 digits of $ 2032^{2031^{2030^{dots^{2^{1}}}}}$












7












$begingroup$


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

So obviously we are looking for $x$ so that
$$
2032^{2031^{2030^{dots^{2^{1}}}}} equiv x quad text{mod}hspace{0.1cm} 1000
$$

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











$endgroup$








  • 2




    $begingroup$
    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
    yesterday










  • $begingroup$
    @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
    yesterday












  • $begingroup$
    Bah, that's no monster. Graham's Number is a monster!
    $endgroup$
    – Cort Ammon
    yesterday
















7












$begingroup$


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

So obviously we are looking for $x$ so that
$$
2032^{2031^{2030^{dots^{2^{1}}}}} equiv x quad text{mod}hspace{0.1cm} 1000
$$

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











$endgroup$








  • 2




    $begingroup$
    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
    yesterday










  • $begingroup$
    @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
    yesterday












  • $begingroup$
    Bah, that's no monster. Graham's Number is a monster!
    $endgroup$
    – Cort Ammon
    yesterday














7












7








7


1



$begingroup$


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

So obviously we are looking for $x$ so that
$$
2032^{2031^{2030^{dots^{2^{1}}}}} equiv x quad text{mod}hspace{0.1cm} 1000
$$

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
$$
2032^{2031^{2030^{dots^{2^{1}}}}} equiv x quad text{mod}hspace{0.1cm} 1000
$$

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








edited yesterday









Asaf Karagila

307k33439770




307k33439770










asked yesterday









Kristin PeterselKristin Petersel

363




363








  • 2




    $begingroup$
    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
    yesterday










  • $begingroup$
    @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
    yesterday












  • $begingroup$
    Bah, that's no monster. Graham's Number is a monster!
    $endgroup$
    – Cort Ammon
    yesterday














  • 2




    $begingroup$
    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
    yesterday










  • $begingroup$
    @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
    yesterday












  • $begingroup$
    Bah, that's no monster. Graham's Number is a monster!
    $endgroup$
    – Cort Ammon
    yesterday








2




2




$begingroup$
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
yesterday




$begingroup$
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
yesterday












$begingroup$
@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
yesterday






$begingroup$
@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
yesterday














$begingroup$
Bah, that's no monster. Graham's Number is a monster!
$endgroup$
– Cort Ammon
yesterday




$begingroup$
Bah, that's no monster. Graham's Number is a monster!
$endgroup$
– Cort Ammon
yesterday










4 Answers
4






active

oldest

votes


















6












$begingroup$

$bmod 1000!: 32^{large 2031^{LARGE 2k}}!!equiv, 8left[dfrac{color{#0a0}{32^{large 2031^{LARGE 2k}}}}8 bmod color{#0a0}{125}right]! equiv 8left[dfrac{color{#c00}{32}}8bmod 125right]! equiv 32, $ by



$ ,begin{align} !bmod color{#0a0}{125}!: color{#0a0}{32^{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{#b6f}{2031}^{LARGE color{#d4f}2k}! bmod 20)} {rm by mod Distributive Law}\
&equiv,{2^{large 5(color{#b6f}1^{LARGE k})}}equiv, color{#c00}{32} {rm by} color{#b6f}{2031^{large 2}}!equiv 11^{large 2}equivcolor{#b6f} 1!!!pmod{!20}\
end{align} $






share|cite|improve this answer











$endgroup$









  • 2




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












  • $begingroup$
    How would you prove this?
    $endgroup$
    – Markus Punnar
    21 hours ago










  • $begingroup$
    @Markus Prove what? If you mean the mod Distributive Law then follow the link in my prior comment.
    $endgroup$
    – Bill Dubuque
    19 hours ago



















4












$begingroup$

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



$2031^{10k}=(2030+1)^{10k}=(text{binomial expansion})=100m+1$



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 $color{blue}{032}$ in the base $2032$!






share|cite|improve this answer











$endgroup$









  • 2




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












  • $begingroup$
    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
    yesterday





















4












$begingroup$

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 $monster = 2031^{littlemonster}equiv 31^{littlemonster}pmod {100}$



$31$ 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 $2032^{monster} equiv 32 pmod {125}$



So $2032^{monster} equiv 0 pmod 8$ and $2032^{monster} equiv 32 pmod {125}$.



As $8|32$ we are done. $2032^{monster} equiv 32 pmod {1000}$.



and the last three digits are $032$.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Good idea, but you rendered $2031^{2030^{···}} bmod 125$. You need $2032^{2031^{2030^{···}}} bmod 125$.
    $endgroup$
    – Oscar Lanzi
    yesterday










  • $begingroup$
    oops..............
    $endgroup$
    – fleablood
    20 hours ago






  • 1




    $begingroup$
    Actually I rendered it as $2032^{2031^{...}}equiv 31^{2031^{...}}pmod{125}$... which is still an error.
    $endgroup$
    – fleablood
    20 hours ago



















2












$begingroup$

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









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3161051%2ffind-last-3-digits-of-203220312030-dots21%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    4 Answers
    4






    active

    oldest

    votes








    4 Answers
    4






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    6












    $begingroup$

    $bmod 1000!: 32^{large 2031^{LARGE 2k}}!!equiv, 8left[dfrac{color{#0a0}{32^{large 2031^{LARGE 2k}}}}8 bmod color{#0a0}{125}right]! equiv 8left[dfrac{color{#c00}{32}}8bmod 125right]! equiv 32, $ by



    $ ,begin{align} !bmod color{#0a0}{125}!: color{#0a0}{32^{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{#b6f}{2031}^{LARGE color{#d4f}2k}! bmod 20)} {rm by mod Distributive Law}\
    &equiv,{2^{large 5(color{#b6f}1^{LARGE k})}}equiv, color{#c00}{32} {rm by} color{#b6f}{2031^{large 2}}!equiv 11^{large 2}equivcolor{#b6f} 1!!!pmod{!20}\
    end{align} $






    share|cite|improve this answer











    $endgroup$









    • 2




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












    • $begingroup$
      How would you prove this?
      $endgroup$
      – Markus Punnar
      21 hours ago










    • $begingroup$
      @Markus Prove what? If you mean the mod Distributive Law then follow the link in my prior comment.
      $endgroup$
      – Bill Dubuque
      19 hours ago
















    6












    $begingroup$

    $bmod 1000!: 32^{large 2031^{LARGE 2k}}!!equiv, 8left[dfrac{color{#0a0}{32^{large 2031^{LARGE 2k}}}}8 bmod color{#0a0}{125}right]! equiv 8left[dfrac{color{#c00}{32}}8bmod 125right]! equiv 32, $ by



    $ ,begin{align} !bmod color{#0a0}{125}!: color{#0a0}{32^{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{#b6f}{2031}^{LARGE color{#d4f}2k}! bmod 20)} {rm by mod Distributive Law}\
    &equiv,{2^{large 5(color{#b6f}1^{LARGE k})}}equiv, color{#c00}{32} {rm by} color{#b6f}{2031^{large 2}}!equiv 11^{large 2}equivcolor{#b6f} 1!!!pmod{!20}\
    end{align} $






    share|cite|improve this answer











    $endgroup$









    • 2




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












    • $begingroup$
      How would you prove this?
      $endgroup$
      – Markus Punnar
      21 hours ago










    • $begingroup$
      @Markus Prove what? If you mean the mod Distributive Law then follow the link in my prior comment.
      $endgroup$
      – Bill Dubuque
      19 hours ago














    6












    6








    6





    $begingroup$

    $bmod 1000!: 32^{large 2031^{LARGE 2k}}!!equiv, 8left[dfrac{color{#0a0}{32^{large 2031^{LARGE 2k}}}}8 bmod color{#0a0}{125}right]! equiv 8left[dfrac{color{#c00}{32}}8bmod 125right]! equiv 32, $ by



    $ ,begin{align} !bmod color{#0a0}{125}!: color{#0a0}{32^{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{#b6f}{2031}^{LARGE color{#d4f}2k}! bmod 20)} {rm by mod Distributive Law}\
    &equiv,{2^{large 5(color{#b6f}1^{LARGE k})}}equiv, color{#c00}{32} {rm by} color{#b6f}{2031^{large 2}}!equiv 11^{large 2}equivcolor{#b6f} 1!!!pmod{!20}\
    end{align} $






    share|cite|improve this answer











    $endgroup$



    $bmod 1000!: 32^{large 2031^{LARGE 2k}}!!equiv, 8left[dfrac{color{#0a0}{32^{large 2031^{LARGE 2k}}}}8 bmod color{#0a0}{125}right]! equiv 8left[dfrac{color{#c00}{32}}8bmod 125right]! equiv 32, $ by



    $ ,begin{align} !bmod color{#0a0}{125}!: color{#0a0}{32^{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{#b6f}{2031}^{LARGE color{#d4f}2k}! bmod 20)} {rm by mod Distributive Law}\
    &equiv,{2^{large 5(color{#b6f}1^{LARGE k})}}equiv, color{#c00}{32} {rm by} color{#b6f}{2031^{large 2}}!equiv 11^{large 2}equivcolor{#b6f} 1!!!pmod{!20}\
    end{align} $







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited yesterday

























    answered yesterday









    Bill DubuqueBill Dubuque

    213k29195654




    213k29195654








    • 2




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












    • $begingroup$
      How would you prove this?
      $endgroup$
      – Markus Punnar
      21 hours ago










    • $begingroup$
      @Markus Prove what? If you mean the mod Distributive Law then follow the link in my prior comment.
      $endgroup$
      – Bill Dubuque
      19 hours ago














    • 2




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












    • $begingroup$
      How would you prove this?
      $endgroup$
      – Markus Punnar
      21 hours ago










    • $begingroup$
      @Markus Prove what? If you mean the mod Distributive Law then follow the link in my prior comment.
      $endgroup$
      – Bill Dubuque
      19 hours ago








    2




    2




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






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














    $begingroup$
    How would you prove this?
    $endgroup$
    – Markus Punnar
    21 hours ago




    $begingroup$
    How would you prove this?
    $endgroup$
    – Markus Punnar
    21 hours ago












    $begingroup$
    @Markus Prove what? If you mean the mod Distributive Law then follow the link in my prior comment.
    $endgroup$
    – Bill Dubuque
    19 hours ago




    $begingroup$
    @Markus Prove what? If you mean the mod Distributive Law then follow the link in my prior comment.
    $endgroup$
    – Bill Dubuque
    19 hours ago











    4












    $begingroup$

    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



    $2031^{10k}=(2030+1)^{10k}=(text{binomial expansion})=100m+1$



    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 $color{blue}{032}$ in the base $2032$!






    share|cite|improve this answer











    $endgroup$









    • 2




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












    • $begingroup$
      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
      yesterday


















    4












    $begingroup$

    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



    $2031^{10k}=(2030+1)^{10k}=(text{binomial expansion})=100m+1$



    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 $color{blue}{032}$ in the base $2032$!






    share|cite|improve this answer











    $endgroup$









    • 2




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












    • $begingroup$
      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
      yesterday
















    4












    4








    4





    $begingroup$

    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



    $2031^{10k}=(2030+1)^{10k}=(text{binomial expansion})=100m+1$



    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 $color{blue}{032}$ 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



    $2031^{10k}=(2030+1)^{10k}=(text{binomial expansion})=100m+1$



    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 $color{blue}{032}$ in the base $2032$!







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited yesterday

























    answered yesterday









    Oscar LanziOscar Lanzi

    13.3k12136




    13.3k12136








    • 2




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












    • $begingroup$
      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
      yesterday
















    • 2




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












    • $begingroup$
      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
      yesterday










    2




    2




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






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














    $begingroup$
    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
    yesterday






    $begingroup$
    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
    yesterday













    4












    $begingroup$

    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 $monster = 2031^{littlemonster}equiv 31^{littlemonster}pmod {100}$



    $31$ 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 $2032^{monster} equiv 32 pmod {125}$



    So $2032^{monster} equiv 0 pmod 8$ and $2032^{monster} equiv 32 pmod {125}$.



    As $8|32$ we are done. $2032^{monster} equiv 32 pmod {1000}$.



    and the last three digits are $032$.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Good idea, but you rendered $2031^{2030^{···}} bmod 125$. You need $2032^{2031^{2030^{···}}} bmod 125$.
      $endgroup$
      – Oscar Lanzi
      yesterday










    • $begingroup$
      oops..............
      $endgroup$
      – fleablood
      20 hours ago






    • 1




      $begingroup$
      Actually I rendered it as $2032^{2031^{...}}equiv 31^{2031^{...}}pmod{125}$... which is still an error.
      $endgroup$
      – fleablood
      20 hours ago
















    4












    $begingroup$

    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 $monster = 2031^{littlemonster}equiv 31^{littlemonster}pmod {100}$



    $31$ 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 $2032^{monster} equiv 32 pmod {125}$



    So $2032^{monster} equiv 0 pmod 8$ and $2032^{monster} equiv 32 pmod {125}$.



    As $8|32$ we are done. $2032^{monster} equiv 32 pmod {1000}$.



    and the last three digits are $032$.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Good idea, but you rendered $2031^{2030^{···}} bmod 125$. You need $2032^{2031^{2030^{···}}} bmod 125$.
      $endgroup$
      – Oscar Lanzi
      yesterday










    • $begingroup$
      oops..............
      $endgroup$
      – fleablood
      20 hours ago






    • 1




      $begingroup$
      Actually I rendered it as $2032^{2031^{...}}equiv 31^{2031^{...}}pmod{125}$... which is still an error.
      $endgroup$
      – fleablood
      20 hours ago














    4












    4








    4





    $begingroup$

    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 $monster = 2031^{littlemonster}equiv 31^{littlemonster}pmod {100}$



    $31$ 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 $2032^{monster} equiv 32 pmod {125}$



    So $2032^{monster} equiv 0 pmod 8$ and $2032^{monster} equiv 32 pmod {125}$.



    As $8|32$ we are done. $2032^{monster} equiv 32 pmod {1000}$.



    and the last three digits are $032$.






    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 $monster = 2031^{littlemonster}equiv 31^{littlemonster}pmod {100}$



    $31$ 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 $2032^{monster} equiv 32 pmod {125}$



    So $2032^{monster} equiv 0 pmod 8$ and $2032^{monster} equiv 32 pmod {125}$.



    As $8|32$ we are done. $2032^{monster} equiv 32 pmod {1000}$.



    and the last three digits are $032$.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 20 hours ago

























    answered yesterday









    fleabloodfleablood

    73.4k22791




    73.4k22791








    • 1




      $begingroup$
      Good idea, but you rendered $2031^{2030^{···}} bmod 125$. You need $2032^{2031^{2030^{···}}} bmod 125$.
      $endgroup$
      – Oscar Lanzi
      yesterday










    • $begingroup$
      oops..............
      $endgroup$
      – fleablood
      20 hours ago






    • 1




      $begingroup$
      Actually I rendered it as $2032^{2031^{...}}equiv 31^{2031^{...}}pmod{125}$... which is still an error.
      $endgroup$
      – fleablood
      20 hours ago














    • 1




      $begingroup$
      Good idea, but you rendered $2031^{2030^{···}} bmod 125$. You need $2032^{2031^{2030^{···}}} bmod 125$.
      $endgroup$
      – Oscar Lanzi
      yesterday










    • $begingroup$
      oops..............
      $endgroup$
      – fleablood
      20 hours ago






    • 1




      $begingroup$
      Actually I rendered it as $2032^{2031^{...}}equiv 31^{2031^{...}}pmod{125}$... which is still an error.
      $endgroup$
      – fleablood
      20 hours ago








    1




    1




    $begingroup$
    Good idea, but you rendered $2031^{2030^{···}} bmod 125$. You need $2032^{2031^{2030^{···}}} bmod 125$.
    $endgroup$
    – Oscar Lanzi
    yesterday




    $begingroup$
    Good idea, but you rendered $2031^{2030^{···}} bmod 125$. You need $2032^{2031^{2030^{···}}} bmod 125$.
    $endgroup$
    – Oscar Lanzi
    yesterday












    $begingroup$
    oops..............
    $endgroup$
    – fleablood
    20 hours ago




    $begingroup$
    oops..............
    $endgroup$
    – fleablood
    20 hours ago




    1




    1




    $begingroup$
    Actually I rendered it as $2032^{2031^{...}}equiv 31^{2031^{...}}pmod{125}$... which is still an error.
    $endgroup$
    – fleablood
    20 hours ago




    $begingroup$
    Actually I rendered it as $2032^{2031^{...}}equiv 31^{2031^{...}}pmod{125}$... which is still an error.
    $endgroup$
    – fleablood
    20 hours ago











    2












    $begingroup$

    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









    $endgroup$


















      2












      $begingroup$

      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









      $endgroup$
















        2












        2








        2





        $begingroup$

        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 yesterday









        IngixIngix

        5,077159




        5,077159






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


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

            But avoid



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

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


            Use MathJax to format equations. MathJax reference.


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




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3161051%2ffind-last-3-digits-of-203220312030-dots21%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Færeyskur hestur Heimild | Tengill | Tilvísanir | LeiðsagnarvalRossið - síða um færeyska hrossið á færeyskuGott ár hjá færeyska hestinum

            He _____ here since 1970 . Answer needed [closed]What does “since he was so high” mean?Meaning of “catch birds for”?How do I ensure “since” takes the meaning I want?“Who cares here” meaningWhat does “right round toward” mean?the time tense (had now been detected)What does the phrase “ring around the roses” mean here?Correct usage of “visited upon”Meaning of “foiled rail sabotage bid”It was the third time I had gone to Rome or It is the third time I had been to Rome

            Slayer Innehåll Historia | Stil, komposition och lyrik | Bandets betydelse och framgångar | Sidoprojekt och samarbeten | Kontroverser | Medlemmar | Utmärkelser och nomineringar | Turnéer och festivaler | Diskografi | Referenser | Externa länkar | Navigeringsmenywww.slayer.net”Metal Massacre vol. 1””Metal Massacre vol. 3””Metal Massacre Volume III””Show No Mercy””Haunting the Chapel””Live Undead””Hell Awaits””Reign in Blood””Reign in Blood””Gold & Platinum – Reign in Blood””Golden Gods Awards Winners”originalet”Kerrang! Hall Of Fame””Slayer Looks Back On 37-Year Career In New Video Series: Part Two””South of Heaven””Gold & Platinum – South of Heaven””Seasons in the Abyss””Gold & Platinum - Seasons in the Abyss””Divine Intervention””Divine Intervention - Release group by Slayer””Gold & Platinum - Divine Intervention””Live Intrusion””Undisputed Attitude””Abolish Government/Superficial Love””Release “Slatanic Slaughter: A Tribute to Slayer” by Various Artists””Diabolus in Musica””Soundtrack to the Apocalypse””God Hates Us All””Systematic - Relationships””War at the Warfield””Gold & Platinum - War at the Warfield””Soundtrack to the Apocalypse””Gold & Platinum - Still Reigning””Metallica, Slayer, Iron Mauden Among Winners At Metal Hammer Awards””Eternal Pyre””Eternal Pyre - Slayer release group””Eternal Pyre””Metal Storm Awards 2006””Kerrang! Hall Of Fame””Slayer Wins 'Best Metal' Grammy Award””Slayer Guitarist Jeff Hanneman Dies””Bullet-For My Valentine booed at Metal Hammer Golden Gods Awards””Unholy Aliance””The End Of Slayer?””Slayer: We Could Thrash Out Two More Albums If We're Fast Enough...””'The Unholy Alliance: Chapter III' UK Dates Added”originalet”Megadeth And Slayer To Co-Headline 'Canadian Carnage' Trek”originalet”World Painted Blood””Release “World Painted Blood” by Slayer””Metallica Heading To Cinemas””Slayer, Megadeth To Join Forces For 'European Carnage' Tour - Dec. 18, 2010”originalet”Slayer's Hanneman Contracts Acute Infection; Band To Bring In Guest Guitarist””Cannibal Corpse's Pat O'Brien Will Step In As Slayer's Guest Guitarist”originalet”Slayer’s Jeff Hanneman Dead at 49””Dave Lombardo Says He Made Only $67,000 In 2011 While Touring With Slayer””Slayer: We Do Not Agree With Dave Lombardo's Substance Or Timeline Of Events””Slayer Welcomes Drummer Paul Bostaph Back To The Fold””Slayer Hope to Unveil Never-Before-Heard Jeff Hanneman Material on Next Album””Slayer Debut New Song 'Implode' During Surprise Golden Gods Appearance””Release group Repentless by Slayer””Repentless - Slayer - Credits””Slayer””Metal Storm Awards 2015””Slayer - to release comic book "Repentless #1"””Slayer To Release 'Repentless' 6.66" Vinyl Box Set””BREAKING NEWS: Slayer Announce Farewell Tour””Slayer Recruit Lamb of God, Anthrax, Behemoth + Testament for Final Tour””Slayer lägger ner efter 37 år””Slayer Announces Second North American Leg Of 'Final' Tour””Final World Tour””Slayer Announces Final European Tour With Lamb of God, Anthrax And Obituary””Slayer To Tour Europe With Lamb of God, Anthrax And Obituary””Slayer To Play 'Last French Show Ever' At Next Year's Hellfst””Slayer's Final World Tour Will Extend Into 2019””Death Angel's Rob Cavestany On Slayer's 'Farewell' Tour: 'Some Of Us Could See This Coming'””Testament Has No Plans To Retire Anytime Soon, Says Chuck Billy””Anthrax's Scott Ian On Slayer's 'Farewell' Tour Plans: 'I Was Surprised And I Wasn't Surprised'””Slayer””Slayer's Morbid Schlock””Review/Rock; For Slayer, the Mania Is the Message””Slayer - Biography””Slayer - Reign In Blood”originalet”Dave Lombardo””An exclusive oral history of Slayer”originalet”Exclusive! Interview With Slayer Guitarist Jeff Hanneman”originalet”Thinking Out Loud: Slayer's Kerry King on hair metal, Satan and being polite””Slayer Lyrics””Slayer - Biography””Most influential artists for extreme metal music””Slayer - Reign in Blood””Slayer guitarist Jeff Hanneman dies aged 49””Slatanic Slaughter: A Tribute to Slayer””Gateway to Hell: A Tribute to Slayer””Covered In Blood””Slayer: The Origins of Thrash in San Francisco, CA.””Why They Rule - #6 Slayer”originalet”Guitar World's 100 Greatest Heavy Metal Guitarists Of All Time”originalet”The fans have spoken: Slayer comes out on top in readers' polls”originalet”Tribute to Jeff Hanneman (1964-2013)””Lamb Of God Frontman: We Sound Like A Slayer Rip-Off””BEHEMOTH Frontman Pays Tribute To SLAYER's JEFF HANNEMAN””Slayer, Hatebreed Doing Double Duty On This Year's Ozzfest””System of a Down””Lacuna Coil’s Andrea Ferro Talks Influences, Skateboarding, Band Origins + More””Slayer - Reign in Blood””Into The Lungs of Hell””Slayer rules - en utställning om fans””Slayer and Their Fans Slashed Through a No-Holds-Barred Night at Gas Monkey””Home””Slayer””Gold & Platinum - The Big 4 Live from Sofia, Bulgaria””Exclusive! Interview With Slayer Guitarist Kerry King””2008-02-23: Wiltern, Los Angeles, CA, USA””Slayer's Kerry King To Perform With Megadeth Tonight! - Oct. 21, 2010”originalet”Dave Lombardo - Biography”Slayer Case DismissedArkiveradUltimate Classic Rock: Slayer guitarist Jeff Hanneman dead at 49.”Slayer: "We could never do any thing like Some Kind Of Monster..."””Cannibal Corpse'S Pat O'Brien Will Step In As Slayer'S Guest Guitarist | The Official Slayer Site”originalet”Slayer Wins 'Best Metal' Grammy Award””Slayer Guitarist Jeff Hanneman Dies””Kerrang! Awards 2006 Blog: Kerrang! Hall Of Fame””Kerrang! Awards 2013: Kerrang! Legend”originalet”Metallica, Slayer, Iron Maien Among Winners At Metal Hammer Awards””Metal Hammer Golden Gods Awards””Bullet For My Valentine Booed At Metal Hammer Golden Gods Awards””Metal Storm Awards 2006””Metal Storm Awards 2015””Slayer's Concert History””Slayer - Relationships””Slayer - Releases”Slayers officiella webbplatsSlayer på MusicBrainzOfficiell webbplatsSlayerSlayerr1373445760000 0001 1540 47353068615-5086262726cb13906545x(data)6033143kn20030215029