Find the multiplicative inverse of 6 in Z20Solution6 does no

Find the multiplicative inverse of 6 in Z₂₀

Solution

6 does not have an inverse in Z20, because the equation 6.20 x = 1 does not have a solution.

(This can be verified by checking the 20 possible values for x.)

If a does have an inverse a , then we can find a solution to the equation

a ·n x = b .

To do so, we multiply both sides of the equation by a ,

obtaining a ·n (a ·n x) = a ·n b. By the associative law,

this gives us (a ·n a) ·n x = b.

But a ·n a = 1 by definition so we have that x = a ·n b . Since this computation is valid for any x that satisfies the equation, we conclude that the only x that satisfies the equation is a ·n b.

in other words the gcd(a.x) > 1 so a does not have multiplicative inverse in Z20.