In Raymond Murphy's "English Grammar in Use with Answers 4th edition" there is the following sentence:
I couldn't get into the house because I didn't have a key.
with "a" being the correct answer in the exercise keys.
Am I correct to understand that "the" would've been more suitable? If not, what is the logic here?
After all, only one specific key (or at least one of its duplicates) will unlock the door.
P.S. I'd like to add that the sentence comes from a unit on using indefinite articles specifically, so the book does not explicitly state that using "the" would've been wrong here. However, it feels strange to me regardless that such a sentence would be included.