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Math Help - units digit of the product (3^75)(2^113)

  1. #1
    Super Member bigwave's Avatar
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    units digit of the product (3^75)(2^113)

    Q1. What is the units digit of the product (3^{75})(2^{113})

    was not sure of what "units digit" meant but this comes out to be

     <br />
6.31  10^{69}<br />

    otherwise how is this found?

    Q2. The product of three different positive intergers is 2010. What is the maximum possible sum of the intergers?
    Last edited by bigwave; January 8th 2011 at 01:43 PM. Reason: added Q2
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  2. #2
    MHF Contributor
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    Solve mod 10

    3^{75}*2^{113}\equiv x \ \mbox{(mod 10)}

    3^4\equiv 1 \ \mbox{(mod 10)}

    3^{72+3}=(3^4)^{18}*3^3\equiv 1^{18}*3^3\equiv 7 \ \mbox{(mod 10)}

    Get the idea?
    Last edited by dwsmith; January 8th 2011 at 01:41 PM.
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    hint for q2:

    2010=2*3*5*67
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    for q1:

    better and easy to compute 6^75(mod10)

    and the we left with: 2^38(mod10)=4^6*2^2=2^14=8^2=4(mod10)
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