I'm not 100% certain my kitchen floor is linoleum, but it came in a roll when they put it on years ago, it's cheap I'm sure and has fake tile looking indents where tile grout would have been.
I never could get mine clean either until one day I just decided that to clean it properly would require getting on my hands and knees with a sponge. Well 3 types of cleaners later and I still couldn't get it clean. I got mad and just dumped straight bleach (regular clorox) on one area, let it set for about 5 minutes, then wiped it up, scrubbing on the stubborn areas. My floor never looked so clean before! I still have to do that about once a year despite mopping and spot cleaning spills though.
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