# Why 1 is not a prime number?

QuestionsCategory: MathematicsWhy 1 is not a prime number?
Raymond John Staff asked 3 weeks ago

You have heard of prime numbers. And according to dictionary, prime number is a number that is divisible by itself and 1, so why number 1 is not prime number?

Note that when we say that a number is “prime”, all that we are doing is applying a definition that was devised by mathematicians. A prime number is generally defined to be any positive number that has exactly two distinct positive integer divisors (the divisors being 1 and the number itself). So 13 is prime, because it is divisible only by 1 and 13, whereas 14 is not prime because it is divisible by 1, 2, 7 and 14. Note that this excludes the number 1 from being prime. The biggest reason this definition of primality is used, as opposed to a slightly different one, is merely a matter of convenience. Mathematicians like to choose definitions in such a way that important theorems are simple and easy to state. Probably the most important theorem involving prime numbers is the Fundamental Theorem of Arithmetic, which says that all integers greater than 1 can be expressed as a unique product of prime numbers up to reordering of the factors. So, for example, 54 can be written as $54 = 3*3*3*2$ which is a unique factorization assuming that we list the factors in decreasing order. Now, notice that if we counted 1 as a prime number, then this theorem would no longer hold as stated, since we would then be able to write
$54 = 3*3*3*2*1 = 3*3*3*2*1*1 = 3*3*3*2*1*1*1$