Thursday, 16 January 2014

number theory - Find minimal xinBbbN that solves the linear congruence



I need to find minimal xN that solves the linear congruences:



6x \equiv 2 \pmod {\!4}




3x \equiv 6 \pmod {\!9}



x \equiv 15 \pmod {\! 17}



I divided the first congruence by 2 and the second congruence by 3, then used the Chinese remainder theorem.



I got x\equiv 393 \pmod {2\cdot 3\cdot 17}



I checked that this solution actually solves the 3 equations.




But is it the smallest one? If so, then why ?



Thanks.


Answer



Your congruences may be reduced to



x\equiv 1\pmod{2}



x\equiv 2 \pmod{3}




x\equiv 15 \pmod{17}



We have \mod {17}, x\equiv 15\equiv 32\equiv 49 \equiv 66 \equiv 83.



And we note that 83 satisfies the other two congruences.



The general solution is then x\equiv 83 \pmod{102}


No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...