If your MTB is 21 speeds I will assume it is a 3 X 7 with a 7 speed freewheel. There are 8 speed freewheels but they're not common. I upgraded my old TREK 820 to a 3 X 9 with an swap to a wheelset with a cassette hub in the rear wheel allowing me to run a 9 speed cassette. Even if your rear wheel is a 7 speed cassette it probably wouldn't accept an 8 or 9 speed cassette. I also upgraded my rear derailleur and changed my shifters to match. I went with Shimano Deore XT but you could use one of their less expensive series. Your existing rear derailleur "might" handle a 9 speed but I wouldn't bet on it.
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