# Why is 1 not a Prime Number?

**Is 1 a Prime Number?**

Friends, in one of the post where I have described ‘Prime’ and ‘Composite’ Numbers, one of the curious visitor have asked me a very logical question. I will quote that question for your reference –

**Text from Previous post-**

“Prime and Composite : Any integer which is divisible by 1 and itself only is called a prime number.

unquote

quote

N.B.: 1 is not a prime number.”

**Question**

Could you explaine what is the creteria thar excludes 1 from the list of prime numbers?

a) 1 is integer

b) 1 is divisible by 1 and itself (1)

Since anybody in the past has declared that 1 is not prime number, why we should follow this without thinking and contravene the general rule for prime numbers?

Is 1 as a figure is something which has come from the thin air. It is and always will be an integer. The criteria for 2 are the same – divisible by 1 and itself. And for all prime numbers.

Most probably the 1 is “guilty” because with 1 starts the series on numbers (natural, odd or prime). Suppose 2 was the beginning of the series. Should we ignore 2, because series starts with 2?

**My Explanation-**

1 can be rejected being a prime number because of the given reasons-

The “real” definition of a prime number is “a natural number that has exactly two distinct natural number divisors.” This definition can be considered little confusing for general masses. This in essence means ” Any integer which is divisible by 1 and itself only is called a prime number.”, which is easier to digest. The only problem is that if one uses that phrasing, the number 1 is a little grey zone case. “Well, it is divisible by 1, and it is divisible by itself,” you could think. “Isn’t it also a prime number then?”

No, not by the official definition, because it only has a single natural number divisor: 1. This is why the “exception” had to be made, that 1 is not a prime number.

In short: the definition as we know it is a simplification that doesn’t work completely – except if we specify that 1 is not included.

Is it really important whether 1 is or not a prime number?

It is indeed very crucial to make the distinction. If we consider 1 not to be a prime number, then any composite number (such as 20) can be written as a product of primes in only one way (here, 2*2*5), not counting different orders. However, if 1 were a prime number, there would be infinitely many ways! We could write 20 for example, as 2*2*5, or 1*2*2*5, or 1*1*1*1*1*2*2*5. Having only one way to write a number as a product of primes is very useful when doing math.

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thanks for the answer.

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very nice explanation that i have not ever seen before.and also thankful to the person who raised that doubt

Great explanation!!

GOOD EXPLANATION !!!!!!!

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Regards

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Pre Anu

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