give a damn

Also, not give a fig or hang or hoot or rap or shit. Not care about, be indifferent to, as in I don't give a damn about him, or She doesn't give a fig if he comes or not. The nouns in all these terms signify something totally worthless. Although probably in oral use for much longer, damn is first recorded in this negative form in the late 1700s and the worthless item it is used to denigrate is a curse. Fig has denoted something small and worthless since about 1400, and hang since the mid-1800s; hoot has been used for the smallest particle since the later 1800s; rap, also for the smallest particle, since the first half of the 1800s, and shit, for excrement, since about 1920. All but the first of these terms are colloquial and the last (using shit) is vulgar.

The verb has one meaning:

Meaning #1: show no concern or interest; always used in the negative
  Synonyms: care a hang, give a hoot