In congruence classes Z/(mZ), reduce the equation a_m*x_m^2=c_m either by finding convenient representation for a_m and b_m or by using the inverse of

hexacordoK 2021-01-30 Answered
In congruence classes ZmZ, reduce the equation amxm2=cm either by finding convenient representation for amandbm or by using the inverse of am. Then find a solution for this congruence directly or by replacing cm : with its appropriate representative in ZmZ. If there is no solution explain why. Here am,bm,xm(=x),cmZmZ:
InZ19Z,[2]x2=[13]:
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gotovub
Answered 2021-01-31 Author has 98 answers

Step 1
Let y=x2 then the given congruence relation becomes,
ayb mod mt i^s2y13mod19
This congruence has unique solution if and only if gcd(a,m)=1
Here gcd(2,19)=1 hence the given congruence has unique solution.
Now since 2 has an inverse, we get y2113mod19 which is the only solution.
The inverse of 2Z19 is 10.
y130mod19y=16
Now y=x216=42. Hence x =4 is the required solution.

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