Positive whole numbers, otherwise known as integers, can
be classified into two types: prime or composite. A number is said to be
prime if it has no other divisor except 1 and itself. For
example, 17 is a prime number since there is no whole number that can
divide it and produce an integer quotient other than 1 and 17.
Otherwise, the number is composite, pertaining to
the fact that it is divisible by positive numbers that are not equal to
one or itself. An instance is the number 24, which is divisible by 2, 3,
4, 6, 8 and 12 aside from 1 and 24.
Note that zero and one are neither prime nor
composite, thus they are known to be special numbers.
Here are some examples:
Example 1: 27
27 is equal to 9 x 3 and 1 x 27. Having four divisors,
it is classified as a composite number.
Example 2: 36
36 is equivalent to 6 x 6, 9 x 4, 18 x 2, 12 x 3 and 36
x 1. Thus it is a composite number.
A good rule to remember is that all even numbers or
multiples of 5 (those digits ending with 5 or 0) are composite, since
they are divisible by 2 and 5 respectively.
Example 3: 23
The number only has two divisors: 23 and 1. Aside from
those, none can be found (unless you consider decimals, but prime and
composite numbers only apple to positive whole numbers). Hence it is a
prime number.
But what if you are given a number so large you cannot
decipher whether it is prime or composite...and you only have two
minutes to answer? This is when the following methods of testing a prime
number comes into the frame.
The Divisibility Test
This test can be done if you are well and truly
acquainted with all the divisibility rules as listed below (but there
are more aside from that!):

A number is
divisible by 2 if the last digit is even.

A number is
divisible by 3 If the sum of the digits is divisible by 3 as
well.

A number is
divisible by 4 if the last two digits form a number divisible
by four.

A number is
divisible by 5 if the last digit of the number is a 5 or a 0.

A number is
divisible by 6 If the number is divisible by both 3 and 2.

Take the
last digit, double it, and subtract it from the rest of the number;
if the answer is divisible by 7 (including 0), then the
number is also.

A number is
divisible by 8 if the last three digits form a number
divisible by eight.

A number is
divisible by 9 if the sum of the digits is also divisible by
9.

A number is
divisible by 10 if the number ends in 0.

Alternately
add and subtract the digits from left to right. If the result is
(including 0) is divisible by 11, the number is also.

A number is
divisible by 12 if the number is divisible by both 3 and 4.

Delete the
last digit from the number, then subtract 9 times the deleted
digit from the remaining number. If what is left is divisible by
13,
then so is the original number.
This test involves only one single step: test whether
all numbers below the given number divides the latter. For example, 13
cannot be divided by any number lower that itself except 1. Thus it is
prime.
Here is another example:
37

Since 2 does not divide 37, it is not divisible by
any multiple of 2.

Since 3 does not divide 37, it is not divisible by
any multiple of 3.

Since 5 does not divide 37, it is not divisible by
any multiple of 5.
The above pattern of nondivisibility goes on until you
reach the number 37, thus it is prime.
However, if you are given a number like 367, this method
becomes impractical, hasty and downright timewasting. This is when you
use the second method.
The Prime
Number Test
This test is relatively more practical since you only
have to get the square root of the number (rounded off) and test all
prime numbers below it for divisibility. If none divides the given
number, it is prime.
Let us work on the example given earlier:
367
The square root of 367 is 19.15724, which can be rounded
off as 19.

367 divided by 19 does not produce an integer value,
so we proceed to the next lower prime number.

367 divided by 17 does not produce an integer value,
so we proceed to the next lower prime number.

367 divided by 13 does not produce an integer value,
so we proceed to the next lower prime number.
This goes on until you reach the lowest prime number 
2. Hence the given number is prime.
