Numbers: Integers
Prime numbers
Each integer can be written in a unique way as a product of special numbers: primes.
Prime
A prime number, or simply a prime, is a natural number that has exactly two natural numbers as divisors: #1# and the number itself.
The word "prime" is also used as an adjective. Instead of "#p# is a prime" you can also say "#p# is prime".
If #m# is an integer and #p# is a prime that divides #m#, then we call #p# a prime divisor of #m#.
A way of writing a natural number as a product of smaller natural numbers is called a factorization. These smaller number are called the factors of the factorization.
The requirement that there are exactly two natural numbers acting as divisors, exclude #1# from being a prime. The number #1# is of course very special, but it is not prime.
The smallest prime number is #2#. This means that all the other even numbers (that is, integer multiples of #2#) are not prime. In other words, an integer is even if and only if #2# is a prime divisor of it.
The one but smallest prime is #3#.
Factorization into primes
Every natural number can be written in exactly one way as a product of prime numbers (in ascending order).
The order of the factors in the product of the theorem is important: #6=2\times 3# and #6=3\times 2# are equivalent. In the first case, the factors #2# and #3# are in ascending order whereas in the second case they are not. By requiring that the order must increase, we exclude #3\times 2#.
Since #12=2^2\times 3# is an abbreviation of #12=2\times2\times 3#, these two factorizations are not considered to be different.
The number #1# seems to be an exception to the rule. By considering #1# to be the empty product, meaning the product with #0# factors, it is no exception.
Another way of phrasing the theorem is: each natural number has a unique factorization (up to ordering) in prime factors (that is, factors that are prime).
Because #24 = 12\times 2#.
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