"Can't do without"
So I was taking a mock exam for the Cambridge certificate and we were doing an exercise where there was an initial phrase and you had to rephrase it with the given word. The phrase was "For me, a mobile phone is a necessity" and the word was "do". I wrote "I can't do without a mobile phone" and my teacher marked it wrong and wrote "I could do with a mobile phone". My answer may have been wrong but I don't think "is a necessity" and "I could do with" are equivalent.
Could someone clarify this for me? Any help is greatly appreciated (: