- #1
parsifal
- 14
- 0
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?
[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?