For the following find all x satisfying the given equation o

For the following, find all x satisfying the given equation or show that no such x exists. 99x equivalence 18 (mod 31). 3x equivalence 6 (mod 9). X^2 + x + 1 equivalence 0 (mod 7).

Solution

a)

99=93+6=6 mod 31

So equation becomes

6x=18 mod 31

6 is coprime to 31 so we can divide both sides by 6. We get

x=3 mod 31

x=31m+3 , m is any integer

b)

3x=6 mod 9

3x-6=9m

x-2=3m

So, x=2 mod 3

x=3m+2, m is any integer

c) x^2+x+1=0 mod 7

1=-6 mod 7

x^2+x-6=0 mod 7

x^2+3x-2x-6=0 mod 7

x(x+3)-2(x+3)=0 mod 7

(x+3)(x-2)=0 mod 7

x=2,-3 mod 7

x=7m+2 or 7n-3

m ,n are any integers