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Math Help - 10^{2001}+1 is divisible by 11

  1. #1
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    10^{2001}+1 is divisible by 11

    10^{2001}+1 is divisible by 11

    \varphi(11)=10\Rightarrow 10^{10}\equiv 1 \ \mbox{(mod 11)}

    10^{10*200+1}+1=(10^{10})^{200}*10+1=1^{200}*10+1\  Rightarrow 11\equiv 0 \ \mbox{(mod 11)}

    Is this correct?
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by dwsmith View Post
    10^{2001}+1 is divisible by 11

    \varphi(11)=10\Rightarrow 10^{10}\equiv 1 \ \mbox{(mod 11)}

    10^{10*200+1}+1=(10^{10})^{200}*10+1=1^{200}*10+1\  Rightarrow 11\equiv 0 \ \mbox{(mod 11)}

    Is this correct?
    Yes this is correct. Another way you could've gone is to apply the divisibility rule for 11.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Correct!
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    10^{2001}+1 is :

    1...(2001  \\ times \\ zero)+1

    or


    1...(2001  \\ times \\ zero)...1

    credit to Mr. chiph588@
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  5. #5
    MHF Contributor alexmahone's Avatar
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    This is how I would do it:

    10^{2001}+1=(11-1)^{2001}+1

    On expanding the RHS with the binomial theorem, we see that it is indeed divisible by 11.
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