The GFC of an odd and even number is always odd True or False?

I think you mean GCF, or greatest common factor, in which case, yes, this is always true. An odd number has no factors of 2 or multiples of 2, so there is no way a factor of an odd number can be divisible by 2. Even if the even number has no odd factors, That would simply mean they have no common factors greater than 1, so the GFC would just be 1.