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)

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)