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 $9$?
$\endgroup$2 Answers
$\begingroup$That is correct, but remember that $-2 \equiv 7 \pmod 9$.
$\endgroup$ $\begingroup$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.
$\endgroup$ 1
