Is the gcf of any two even numbers always even?

Yes

the gcf can be found by factoring the 2 numbers and finding the factors common to both

any even number can be represented as 2n where n is an integer

so 2 even numbers are 2n and 2m

their factored form will be 2*n and 2*m, currently, their greatest common factor is 2*c where c is the gcf of n and m

and the gcf has to be an integer so c is an integer and therfor the gcf of 2n and 2m is even because it is 2c

yes, it is always even

Yes i believe so, the gcf of any two numbers is always even