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Find the remainder of the equation

  1. Jan 24, 2007 #1
    The task is to find the remainder of the equation:
    [tex]\frac{18^2+2^{100}}{11}[/tex]

    Now I know that if
    [tex]a \equiv b\ (mod\ m),\ c \equiv d\ (mod\ m) \Rightarrow[/tex]
    [tex]a + c \equiv b +d\ (mod\ m)[/tex] and [tex]ac \equiv bd\ (mod\ m)[/tex]

    so

    [tex]18^2 \equiv b\ (mod\ 11) \Rightarrow \frac{18^2}{11}=29.454545... \Rightarrow b=18^2-11\cdot 29=5[/tex]
    and d<6 as the remainder b+d < 11.

    But as 2^100 is so large, I can't find d the way I found b. How to find it, or is there some other more convenient way that doesn't involve separating 18^2 and 2^100?
     
  2. jcsd
  3. Jan 24, 2007 #2

    mjsd

    User Avatar
    Homework Helper

    have you tried using this [tex]a\equiv b (\text{mod}\; m) \Rightarrow a^k\equiv b^k (\text{mod}\; m)[/tex] to help?

    The answer should be obvious after the use of this theorem
     
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