05AB1E, 29 bytes
тLʒ"Fizz"D'ÒÖ™DŠ«)¬ÑD5ås3å«è,
To my surprise (or my incapability to use the search function) there was no 05AB1E answer to this question.
###Explanation### тLʒ"Fizz"D'ÒÖ™DŠ«)¬ÑD5ås3å«è, тL # push [1,...,100] ʒ # for each... "Fizz" # Push Fizz (didn't find a way to shorten this one sadly) D # Duplicate 'ÒÖ™ # Push Buzz D # Duplicate Š # Move top item on the stack two slots down « # Concatenate the top items (Results in FizzBuzz) ) # Wrap stack to array ¬Ñ # Get Divisors of N D5å # Does 5 divide it? s3å # Does 3 divide it? « # Concatenate top two items è, # Gets item in the array at the index of the concatenated divisors (indexing wraps around) and prints