The general rule is that you're eligible for an "other state credit" whenever you have income that is taxed by two states. You would claim the credit on your resident state's return.
If you literally moved to DC (as opposed to going there for temporary purposes such as school, while maintaining Utah as your permanent home), then you will file one non-resident return (MD) plus two part-year-resident returns (UT and DC). You would pro-rate your other state credit between UT and DC.
If you maintained your permanent home in UT all year, you need not file a DC tax return since you lived there for fewer than 183 days. In that case you'd file a resident UT return and a non-resident MD return only.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
