Is this really the way to say you need to write down someone's phone number? You can just say seu telefone, instead of seu numero de telefone?
Escrever is simply to write, and anotar is to put down in writing. So anotar works better...
Yes, you can use "escrever" when you're talking to someone, but you should prefer to use "anotar".
But in English you never say write down your phone... you say write down your number. Yes, I reported it... again.
Sorry if a sentence ends in seu telefone and could end in seu numero de telephone or seu numero and someone translates that as your telephone it should be accepted.
The verb is "write down" in the exercise. Perhaps we have different "pages". I agree that "note down" would be unusual.