I have this problem, I need to find the smallest possible solution
x≡3(mod10),x≡11(mod13),x≡15(mod17). x≡3(mod10),x≡11(mod13),x≡15(mod17). I used Chinese remainder theorem and found that: x1=3,x2=11 , and x3=−45??
The solutions would be congruent to v=(13×17)×3+(10×17)×11+(10×13)×(−45)=−3317(mod10×13×17)v=(13×17)×3+(10×17)×11+(10×13)×(−45)=−3317(mod10×13×17). Is that the smaller solution? Is x3=−45x3=−45 correct?
My question is how I can reduce -3317 to have a positive answers.