Answer: C. grandmother
Explanation: The only preposition in this sentence is "with," which can only take nouns after it as objects, so "brother" is discarded because its position in the sentence, as is "lived," which is not even a noun but a verb. The whole object of the preposition "with" is the noun phrase "their grandmother," but the right answer is "grandmother," because "their" (possessive pronoun acting as an adjective) is only a modifier of it.
