The primary working President's residence is the Senate building (also known as 1st building) in the Moscow Kremlin complex. Also the President can use the Grand Kremlin Palace (used for official ceremonies and meetings). Earlier, the President also could use the socalled 14th Administrative Corpus Building (the reserve residence), but in 2016 it was demolished.

What's the best answer to this question: does the russian president live in the kremlin?
yes