British prime minister - article

Good day,

At a conference a journalist asks the British prime minister a rude question. Someone tells him:
"Are you crazy? You're talking to the British prime minister!"
I think that "the" is appropriate because there is only one British prime minister.

However,
"Are you crazy? You're talking to the/a prime minister!"
Here I think it's optional. There are many prime ministers in the world, so "a" works as well as "the".

Do you agree?