Someone else might be able to tell you how to clean grout, but the grout in my bathroom floor was discolored, so I purchased some grout paint at Home Depot. You use a small brush to paint it on (you don't have to avoid getting it on the tiles), then immediately use a dry cloth to wipe over it, wiping the tiles, not the grout. It was fast and easy and worked great. The paint can also be used on kitchen countertops and walls and in shower stalls. It is very durable and will hold up to frequent cleaning. It comes in several colors.
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