"The greatest/largest number of...." is correct, however you'd be understood well in either case.
"The most" indicates the superlative degree for certain adjectives; for example: "the most active; the most intelligent; the most energetic; etc."
"The most" can also indicate the largest amount or number of something for countable nouns: for example: "the most apples; the most points; the most participants; etc."
I hope this explanation helps. If you have any more questions, please let me know.