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Math Help - lest significant digit

  1. #1
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    lest significant digit

    Hi I was wondering if somebody could tell me how to find the least significant decimal digit of a large number - like how would i step through it and is there and algorithm numbers like this in general
    eg 1002^3755
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  2. #2
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    Quote Originally Posted by pablo26 View Post
    Hi I was wondering if somebody could tell me how to find the least significant decimal digit of a large number - like how would i step through it and is there and algorithm numbers like this in general
    eg 1002^3755
    The least significant digit of N is N \pmod {10}.

    So in the case of your example:

    1002^{3755}\equiv 2^{3755} \pmod{10}

    Now (you should know this) 2^{10}=1024
    so:

     <br />
2^{3755}=(2^{10})^{375}2^5\equiv 4^{375} \times 2 \pmod{10}<br />

     <br />
4^{375} \times 2=2^{751}=(2^{10})^{75} \times 2 \equiv 4^{75} \times 2 \pmod{10}<br />

     <br />
4^{75} \times 2=2^{151}=(2^{10})^{15} \times 2 \equiv 4^{15} \times 2 \pmod{10}<br />

     <br />
4^{15} \times 2=2^{31}=(2^{10})^3 \times 2\equiv 4^3 \times 2 \pmod{10}<br />

    and:

     <br />
4^3 \times 2=128 \equiv 8 \pmod{10}<br />

    Hence:

    1002^{3755}\equiv 8 \pmod{10}

    and so the least significant digit of 1002^{3755} is 8.

    .
    Last edited by Constatine11; January 8th 2009 at 02:02 AM.
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