Is 7 a prime number?

Posted on01 November 2021

Number seven is a very interesting number that appears in many places! There are 7 days in a week, 7 wonders of the world, 7 continents, and 7 colors in the rainbow, just to mention some.

In mathematics, 7 is also a very special number that has amazing particular properties. One of them is that 7 is a prime number, as we will verify next.

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, 14 = 2 × 7 = 1 × 14; thus, 1, 2, 7 and 14 are all factors of 14, but only 2 and 7 are proper factors of 14.

A natural number is called prime number if it is greater than 1, and it doesn’t have proper factors. For example, the prime numbers less than 7 are 2, 3, and 5.

Why is 7 a prime number?

Seven is a prime number because it doesn’t have proper factors. In other words, the only factors of 7 are 1 and itself. To be sure of this, let’s verify that none of the numbers greater than 1 and less than 7 divides 7.

The numbers greater than 1 and less than 7 are 2, 3, 4, 5, and 6.

Since 7 is not an even number, neither of 2, 4 or 6 divides 7.
The remainder of dividing 7 by 3 is 1: 7 = (3×2) + 1. Thus, 3 is not a factor of 7.
The remainder of dividing 7 by 5 is 2: 7 = (5×1 ) + 2. Thus, 5 is not a factor of 7.

Therefore, 7 can only be written as the product 1 × 7 = 7. Meaning that its only factors are 1 and itself. Thus, 7 is a prime number.

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 7 is a prime number:

If we split 7 stars into two parts, each part having more than one star, then one part will have more stars than the other. They won’t be equal!
If we split 7 stars into three parts, each part having more than one star, then again one part will have more stars than the others.
If we split 7 stars into four or more parts, one of the parts will have a single star.
The only way of splitting 7 stars into equal parts is putting one star in each of 7 parts. This means that 7 is prime!

Example: Is 27 a prime number?

To answer this question, we should verify if 27 has proper factors. We know that 27 = 3 × 9. Thus, 27 has two proper factors: 3 and 9. This means that 27 is not a prime number!

Also, notice that 27 can be split into three equal parts, each having more than one element. This can only be done because 27 is not a prime number.

Which class of prime number is 7?

There are many different classes of prime numbers. We will name three of them here, and then we will see to which of these classes number 7 belongs.

Classes of Prime Numbers
Primoral prime	It is a prime number of the form $\operatorname{𝑝}=(\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n)+1,\;or\;\operatorname{𝑝}=(\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n)-1$ where $\operatorname{𝑝}_1\times\;\operatorname{𝑝}_2\times\;\operatorname{𝑝}_3\times\dots\times\;\operatorname{𝑝}_n$ are the first n prime numbers.	Yes
Mersenne prime	It is a prime number of the form $\operatorname{𝑝}\;=\;2_n\;-\;1$ where n is an integer.	Yes
Safe prime	It is a prime number of the form 2p+1 where p is also a prime number.	Yes

The amazing 7 is the only prime number belonging to the three of these classes. Let’s find out why:

Number 7 is a primoral prime, because 7 = (2 × 3) + 1, and 2 and 3 are the first two prime numbers.
Number 7 is a Mersenne prime because $7\;=\;2^3–\;1$ . But there’s more! The exponent in the expression $7\;=\;2^3–\;1$ is 3, and 3 is also a Mersenne prime: $3\;=\;2^2\;–\;1.$
Number 7 is a safe prime, because 7 = 2(3) + 1 and 3 is prime.