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Math Help - Find the remainder?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find the remainder?

    Find the remainder which is obtained when the number 10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}} is divided by 7.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    Find the remainder which is obtained when the number 10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}} is divided by 7.
    By fermat's little theorem we know that

    10^6 \equiv 1 \mod (7)

    so 10^{10} = 10^6\cdot 10^4=10^4=1000

    By long divison this is  4 \mod 7

    so 10^{10}=4\mod(7)

    and note that

    10 = 4 \mod (6)
    100 = 4 \mod (6)
    1000 = 4 \mod (6)

    ect ...

    so all of the exponents are 4 mod 6

    so with these two facts we get

    10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4=
    but
    10^4=4 \mod (7)
    10(4)=40

    now mod 7 we get 5
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  3. #3
    Super Member fardeen_gen's Avatar
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    Hello TheEmptySet,
    Thank you! Is it possible to do this by binomial theorem too?
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by fardeen_gen View Post
    Hello TheEmptySet,
    Thank you! Is it possible to do this by binomial theorem too?
    I don't see a way, but you never know.
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