Question

# Solve the linear congruence 7x -= 13(mod 19)

Congruence
Solve the linear congruence
$$\displaystyle{7}{x}\equiv{13}{\left(\text{mod}{19}\right)}$$

2021-02-20
Calculation:
Since $$\displaystyle{7}{x}\equiv{13}{\left(\text{mod}{19}\right)}$$
In order to find an inverse of a modulo m, we look for a multiple of a that exceeds multiple of m by 1.
11.7=1(mod 19)
So, 11 is the inverse of 7 modulo 19.
Multiplying both sides by 11 shows that
11.7x = 11.13(mod 19)
so, 77 = 1(mod 19) and 143 = 77(mod 19)