The GCF of an odd and even number is never even.
The GCF of 2 and any even number is 2.
The GCF of any set of odd numbers is odd because odd numbers don't have any even factors.
By definition, every even number has 2 as a factor. That means that every set of even numbers will have at least 2 as a common factor. The GCF might be higher, but if it has 2 as a factor, it's even.
There's a lot of possibilities. Any set of consecutive even numbers has a GCF of 2.
Any consecutive even numbers have a GCF of 2.
The GCF of any two odd numbers is always odd. The GCF of any two even numbers is always even. The GCF of an odd and an even number is always odd.
All even numbers have two as a factor. Any two even numbers have (at the least) two as a common factor. That means the GCF of any two even numbers will have two as a factor. Any number that has two as a factor is even.
The GCF of any two odd numbers is always odd because odd numbers don't have any even factors. The GCF of any two even numbers is always even because even numbers are divisible by two and any common factors would have at least one two in common. The GCF of an even and an odd number is odd because odd numbers don't have any even factors.
The GCF of an odd and even number is never even.
2 and any even number, and many more
2 and any other even number.
The GCF of 2 and any even number is 2.
Because odd numbers don't have any even factors so they can't have any in common.
Any consecutive even numbers.
If x is an odd number that is a common factor of 2 even numbers, so is 2x. So the answer is yes.
It is impossible to list the GCF of every pair of numbers, because there are infinitely many numbers. If you are looking for the GCF of any number by itself, the GCF of a single number is that number.