I understand we are trying to identify larger primes, say at least more than 20. Further, let us try to divide the number only with prime numbers, as in case they are divisible by a composite number, they will be divisible by its prime factors too.
One of simplest thing that comes to one, who is trying to identify prime numbers, is that a prime number does not have in unit's digit {0,2,4,5,6,8}, as the number will then will be divisible by 2 and 5. Also sum of all the digits should not be divisible by 3. These too themselves will remove a large number of composites.
Another important thing is that one need not try all the primes (other than {2,3,5}, which we have already eliminated).
If the number is n and the prime number just below its square root is m, then we should try only till m. The reason is that if a prime number up to less than m does not divide n, then no other than prime will divide it.
As if n has a factor greater than m, say it is x and other factor is y i.e. x*y=n, then y=n/x<m.
Even for trying to divide by a somewhat large number, one could check using a calculator and if quotient is in decimal fraction, move to next prime number.
