Primes have no factors.............so, how could u eliminate numbers with factors......up the number line????


This type of factorization is sometimes called wheel factorization.  It requires more divisions (because some of the divisors will be composite), but does not require us to have a list of primes available.
Suppose n has twenty-five or more digits, then it is impractical to divide by the primes less than its square root.  If n has two hundred digits, then trial division is impossible--so we need much faster tests.  We discuss several such tests below.