ABSTRACT ALGEBRA problem Need help fast Solve 5x equivalence

ABSTRACT ALGEBRA problem. Need help fast!

Solution

here gcd(5,23) = 1

as given we have,

5x 1 (mod 23)

means

5x = 23k + 1 for some k

This means that there should be value of k in {0,1,2,....,22} such that 5x leaves the reminder of 1 when divided by 23.

hence x = 14 satisfies this condition

if we put x = 14 than 5x = 70

when divided by 23 it leaves a reminder of 1

so we can write

5x 70 1 (mod 23)

hence we say that,

x 14 (mod 23)