2 5 points Find all solution s modulo 13 to the quadratic eq

2. (5 points) Find all solution (s) modulo 13 to the quadratic equation X2 1

We need to find x so that

x^2=-1 mod 13

We do it by taking all cases.

Note we need not worry about all cases ie from 0 to 12 because if a is a solution then so is 13-a

because (13-a)^2=13^2-26a+a^2=a^2 mod 13

So we need to check from 1 to 6 only

1^2=1

2^2=4

3^2=9

4^2=16=3

5^2=25=-1

6^2=36=-3

Hence, x=5 is a solution and hence so is: 13-5=8

So two solutions: x=5,8