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:



6x2(mod4)




3x6(mod9)



x15(mod17)



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



I got x393(mod2317)



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



x1(mod2)



x2(mod3)




x15(mod17)



We have mod17, x1532496683.



And we note that 83 satisfies the other two congruences.



The general solution is then x83(mod102)


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