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Number one is a very important number in mathematics. It is the first positive integer, it divides any integer , any number multiplied by 1 remains the same, and it is its own square and its own root.
In what follows, we are going to discuss a single property the fact that 1 is the only positive integer that is neither prime nor composite. Let’s find out why!
We will start recalling some definitions that we have widely discussed in our Prime Numbers article. If you still feel unfamiliar with these notions, we invite you to first read that article and come back here later.
A factor of a natural number is a positive divisor of the number. A proper factor of a natural number is a factor that is different from 1 and from the number itself. For example, 15 = 3 × 5 = 1 × 15; thus, 1, 3, 5, and 15 are all factors of 15, but only 3 and 5 are proper factors of 15.
A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the first four prime numbers are: 2, 3, 5, and 7.
The first thing to notice is that the only factor of 1 is 1 itself. Thus, 1 doesn’t have proper factors, and therefore it is not a composite number.
On the other hand, the definition of prime number requires the number to be greater than 1. Since 1 is not greater than 1, it doesn’t fit in the definition of a prime number.
In fact, we define prime numbers as greater than 1 exactly to exclude 1. This is what in mathematics is called a “convention” and it is made to be consistent with other definitions.
In particular, it is very important that 1 is not prime to guarantee the validity of the Fundamental Theorem of Arithmetic.
Recall that the Fundamental Theorem of Arithmetic ensures that any integer greater than 1 can be written, in a unique way, as a product of prime numbers.
For example, 18 = 2 × 3 × 3 is the unique representation of 18 as a product of primes.
But, if there were the case that 1 was a prime number, then 18 = 1 × 2 × 3 × 3 and 18 = 1 × 1 × 1 × 2 × 3 × 3, for example, would also be representations of 18 as a product of prime numbers; losing the unicity in the theorem.
So, this is one of the main reasons why 1 is not considered a prime number.
Even more, notice that 1 is different from any prime number, and from any composite number, in the following sense:
Since 1 doesn’t have two different factors, it doesn’t behave as a prime.
Moreover, since 1 doesn’t have more than two factors it doesn’t behave as a composite number. Thus, 1 is the only positive integer that is neither prime nor composite!
We list below some numbers that have 1 as one of its digits. This way, you can have them at hand when deciding if some numbers related to 1 are primes.
Example: Which of the numbers 11, 111, and 1111 is prime?
If we look in the table, we see that 11 is one of the primes ending in 1, and also has two digits equal to 1. So, it is a prime number. You can find additional details on why 11 is a prime number in our article.
Since 111 and 1111 aren’t in the table, we guess that they are not primes. Nevertheless, it isn’t a complete argument, because our table only contains the first few prime numbers with two digits equal to 1, and 1111 is greater than those numbers in the table.
For a complete argument on why 111 and 1111 are not prime numbers, we should verify that they have proper factors. Notice that:
111 = 3 × 371111 = 11 × 101
Thus, 3 and 37 are proper factors of 111; and 11 and 101 are proper factors of 1111. This means that 111 and 1111 are composite numbers: they are not primes. By the way, the two proper factors of 1111 are prime numbers, as our table shows.
Why 101 is a prime number you can read in our next article.
No, by definition, any prime number is greater than 1.
No, because it doesn’t have proper factors.
The first prime number is 2.
1 is the only positive integer having a single factor, which is 1 itself.
Any prime number has two different factors: 1 and itself. Number 1 has only one factor: 1.
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