# 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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#### Vineet Patawari

Hi, I'm Vineet Patawari. I fell in love with numbers after being scared of them for quite some time. Now, I'm here to make you feel comfortable with numbers and help you get rid of Math Phobia!

## 8 thoughts to “Why is 1 not a Prime Number?”

1. divya says:

thanks for the answer.
great answer and great job

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

3. Paul says:

Great explanation!!

4. YOGI says:

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6. Preethi Anusha says:

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Pre Anu

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