    Sentence 1: The sieve of Eratosthenes  would not be valid if what were true?    Sentence 2: Similarly, the sieve of  Eratosthenes would not work correctly  if 1 were considered a prime: a modified  version of the sieve that considers 1 as  prime would eliminate all multiples of  1 (that is, all other numbers) and  produce as output only the single  number 1.