if , then .
This false and only true if , correct?
No, my 3 should be a 3. As it happens, 4 also does not have a multiplicative inverse mod 6. In general, "a" has a multiplicative inverse mod n if and only if gcd(a,n)=1.
Edit: Sorry got a bit confused with everyone posting at the same time, thought the OP was responding to my post. Carry on.