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Thread: Modulo 20 exponents?

  1. #1
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    Modulo 20 exponents?

    Im trying to find 2018^2018 mod 20. Are there rules to help with this? I know a^b = c^b mod 20 if a=c mod 20, so you can do 18^2018 mod 20, but I dont know where to go from there.
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  2. #2
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    Re: Modulo 20 exponents?

    Quote Originally Posted by Ilikebugs View Post
    Im trying to find 2018^2018 mod 20. Are there rules to help with this? I know a^b = c^b mod 20 if a=c mod 20, so you can do 18^2018 mod 20, but I dont know where to go from there.
    For quick answers I go here
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  3. #3
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    Re: Modulo 20 exponents?

    Is there a way to prove the answer without using a calculator or online resource using rules for modular arithmetic?
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  4. #4
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    Re: Modulo 20 exponents?

    $$20=2^2\cdot 5$$

    $$\phi(20)=8$$

    This means:

    $$\forall x,r \in \mathbb{Z}, r>0 , x^{8k+r} \equiv x^r \pmod{20}$$

    So you have:

    $$2018 \equiv 18 \pmod{20}$$

    $$2018 \equiv 2\pmod{8}$$

    So, this gives:

    $$2018^{2018} \equiv 18^2 \pmod{20}$$
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  5. #5
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    Re: Modulo 20 exponents?

    Quote Originally Posted by SlipEternal View Post

    This means:

    $$\forall x,r \in \mathbb{Z}, r>0 , x^{8k+r} \equiv x^r \pmod{20}$$
    Not true

    Take for example $$x=2,k=1,r=1$$
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  6. #6
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    Re: Modulo 20 exponents?

    Quote Originally Posted by Idea View Post
    Not true

    Take for example $$x=2,k=1,r=1$$
    It was late and I had a typo. I meant $r>1$
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