Is 11 a prime number?

In this article, we will be discussing why 11 is a prime number. In fact, it is a special one.

Number 11 is not only the smallest natural number made of a repeated digit, it is also the smallest two-digit prime.

Get ready to learn more properties about 11!    

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, 111 = 3×37=1×111; thus, 1, 3, 37, and 111 are all factors of 111, but only 3 and 37 are proper factors of 111.

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 prime numbers are 2, 3, 5, and 7.

A composite number is a natural number that has proper factors. As we saw, 111 has two proper factors, thus 111 is a composite number.

Why is 11 a prime number?

Number 11 is a prime number because it doesn’t have proper factors. In other words, the only factors of 11 are 1 and itself.

To be sure of it, we can use the following property.

If 𝒏 is a natural number, and neither of the prime numbers is less than $$\sqrt{\mathbf n}$$ divides 𝒏, then 𝒏 is a prime number.

Notice that 11 < 16, thus $$\sqrt{11}<\;\sqrt6\;=\;4$$.

Therefore, the prime numbers less than $$\sqrt{11}$$ are 2 and 3.

Since 11 isn’t even, 2 doesn’t divide 11. Moreover, 11=(3×3)+2, meaning that 3 doesn’t divide 11 either. Then, by the property above, 11 is a prime number.
Most specifically, 11 is the fifth prime number, preceded by 2, 3, 5, and 7.

On the other hand, prime numbers cannot be split into equal parts, each having more than one element.

This is another way of verifying that 11 is a prime number:

• For example, if we split 11 stars into two parts, then one part will have more stars than the other. The parts won’t be equal!
  The same happens if we split 11 stars into three, four, or five parts.
• If we split 11 stars into six or more parts, one of the parts will have a single star.

Thus, the only way of splitting 11 stars into equal parts is putting one star in each of 11 parts. This means that 11 is a prime number!

Which class of prime number is 11?

Since 11 it is the 5th prime number, and 5 is also prime, 11 belongs to the class of super-prime numbers: they are the prime numbers that occupy a prime position in the list of all prime numbers.

Since 11 = 13 – 2 and 13 is another prime, 11 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Number 11 can be classified into many different classes of prime numbers. We will name three more classes here, and then we will see to which of these classes number 11 belongs.

Classes of Prime Numbers
Primoral prime	It is a prime number of the form   $$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)+ 1 $$, or $$\\p=\left(p_1\times p_2\times p_3\times…\times p_n\right)- 1 $$  where $$p_{1,\;\;}p_{2,\;\;}p_{3,\;…,}\;p_n\;$$  are the first n prime numbers.	No
Mersenne prime	It is a prime number of the form   $$p=2^n-1$$ where n is an integer.	No
Safe prime	It is a prime number of the form  2p+1  where p is also a prime number.	Yes

Let’s find out why:

• If we use the first two prime numbers 2 and 3, in the primoral formula,
  we get (2 × 3) − 1 = 5, and (2 × 3) + 1 = 7.
  Moreover, using the first three prime numbers 2, 3, and 5, we get (2 × 3 × 5) − 1 = 29,
  and (2 × 3 × 5) + 1 = 31. Notice that the first two of the resulting numbers are less than 11, and the last two are greater than 11, thus 11 has not the form of a primoral prime.
• Notice that:
  $$2^3-1=8-1=7\\2^4-1=16-1=15$$
  Therefore, 11 is not of the form $$2^n\;–\;1$$, with 𝑛 an integer.
• Notice that 11=2(5)+1, and 5 is a prime number. Thus, 11 is a safe prime.

We invite you to read other articles on prime numbers, on our blog, to find out which other prime numbers belong to these classes.

Do you know that 13 is another prime?