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Math Help - Big power

  1. #1
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    Big power

    I am trying to find a way to obtain the last 3 digits of k,
    where k = 4321^(10^4321 + 1).

    I can see that 10^4321 + 1 = log(k) / log(4321),
    or log(k) / log(4321) - 10^4321 = 1

    However, numbers are so huge here that I think the use of
    some trick or theory is required.

    Can anyone help. Thank you.
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  2. #2
    Eater of Worlds
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    Even though 10^{4321}+1 is a huge number, it still ends in 001.

    Therefore, if I am not mistaken, 4321^{10^{4321}+1} will end in 321.
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  3. #3
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    Hello, Wilmer!

    Find the last 3 digits of k, where: . k \:= \:4321^{10^{4321} + 1}
    I used a very primtive approach . . .

    Since we are concerned with the last three digits only,
    . . we can raise 321 to various powers.

    I found that: . 321^{25} ends in 001.

    That is, consecutive powers of 321 end in: . 321,\,041,\,161,\,681,\,601,\,\cdots \,001
    . . It goes through a 25-step cycle and starts over.


    The power 10^{4321} + 1 \:=\:10^2\!\cdot\!10^{4319} + 1 \:=\:25\!\cdot\!4\!\cdot\!10^{4319} + 1

    \text{Then we have: }\;321^{25\cdot4\cdot10^{4319} + 1} \:=\;321^{25\cdot4\cdot10^{4319}}\cdot321^1 \:=\:\underbrace{(321^{25})}_{\text{This ends in 001}}\,\!\!\!\!\!\!\!^{4\cdot10^{4319}} \cdot321

    So we have: . (001)^{4\cdot10^{4319}}\cdot321 \quad\Rightarrow\quad (001)\cdot 321 \quad\Rightarrow\quad321

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  4. #4
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    What we want is 4321^(10^(4321)+1) mod 1000

    4321^{10^{4321}+1}  \equiv 321^{10^{4321}+1}

    We need to use Euler's totient theorem a^{\phi(n)}\equiv1 mod n

    \phi(1000) is the number of numbers smaller than 1000 and coprime with 1000.
    1000 = 2^35^3
    so we just need to count the number of multiples of 2 and 5 less than 1000. There are 1000/2 = 500 even numbers, 1000/5 = 200 multiples of 5 and 1000/10=100 multiples of 2 and 5.
    This gives us \phi(1000) = 1000-500-200+100 = 400

    so we have
    321^{10^{4321}+1} \equiv 321^{10^{4321}+1 \pmod{400}}
    but 400| 10^4321
    so 321^{10^{4321}+1} \equiv 321^1 =321
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  5. #5
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    Thank you very much everyone.
    I sure learned a lot.
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