# Pumping Lemma for regular language [duplicate]

I have a question to find out that L = {a^(2k)|k>=1} is regular. I know that it is regular set but I was looking to find out if pumping lemma is satisfying or not. So I tried it as -

Let # of states in the FA be n & I select the string w = a^2n
Now let xyz = w
xy = a^n
y = a^m | m is odd number
now x(y^2)z = a^(n-m)(a^2m)(a^n)
= a(2n+m) does not belong to L


which says this regular set doesn't satisfy pumping property. Please let me know if I am doing something wrongly.

Note that the statement is that every string has at least one way of dividing it up. Not every division of a string necessarily works. As you have shown, some don't work, however there is a different division (any one where $m$ is even) that does work. So the language does satisfy the pumping lemma.