Multiplicative inverse question

If I have the additive group (Z_65536,+) (i.e. integers between 0 and 65535 together with addition modulo 65536).

How does Z_65536 does *not* form a group together with multiplication modulo 65536? Someone told me that no all integers have multiplicative inverses.

They tod me that 3 does have a multiplicative inverse in Z_65536, and it is in fact 43691. How do they get this number? Then then told me that 4 doesn't have a multiplicative inverse in Z_65536.

How did they do these calculuations?