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Thread: Need help with this congruency problem

  1. December 13th 2009, 11:03 AM #1
    steph3824
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    Question Need help with this congruency problem

    Show (23^(37)+15^(16)) is congurent to 5 (mod 18).

    Not sure how to do this need some help please!!
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  2. December 13th 2009, 12:46 PM #2
    tonio
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    Quote Originally Posted by steph3824 View Post
    Show (23^(37)+15^(16)) is congurent to 5 (mod 18).

    Not sure how to do this need some help please!!

    I did it two ways and I get something different. The following is done modulo 18:

    23^{37}+15^{16}=5^{37}+5^{16}\cdot 3^{16}=5^{16}\left(5^{21}+3^{16}\right)=\left(5^3\ right)^5\cdot 5\left(\left(5^3\right)^5+\left(3^4\right)^4\right ) =(-1)^5\cdot 5\left((-1)^5+9\right)=(-5)\left((-1)+9\right)=(-5)\cdot 8=-40=-4=14\!\!\!\!\pmod{18}
    Of course, I could be wrong...

    Tonio
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  3. December 13th 2009, 03:03 PM #3
    aidan
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    Quote Originally Posted by steph3824 View Post
    Show (23^(37)+15^(16)) is congurent to 5 (mod 18).

    Not sure how to do this need some help please!!
    23^{37} \equiv 5 \mod 18

    15^{16} \equiv 9 \mod 18

    & 5+9 \equiv 14 \mod 18

    That agrees with tonio.
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