The inverse of $4$ modulo $9$

Question

Can someone explain why the inverse $4$ modulo $9$ is $7$? What am I missing? $$9 = 2\cdot4 + 1$$ $$1 = 9-4\cdot2$$ $$1 = -2\cdot4 + 1\cdot9$$

Isn't then $-2$ inverse of $4$ modulo $9$?

Answer 1

That is correct, but remember that $-2 \equiv 7 \pmod 9$.

Answer 2

To find the modular multiplicative inverse x of 4(mod 9), we must solve the equation

x = 4^(-1)(mod 9)

4x = 1(mod 9)

4x = 28(mod 9)

x = 7(mod 9)

The multiplicative inverse is 7. If you multiply a number by its inverse, you get 1.

7*4 = 28, but 28(mod 9) = 1.