# Is 97 a prime number?

What do the numbers** 900007, 90007, 9007, 907** and **97** have all in common? Firstly, each of them

is obtained by deleting a zero from the previous one. Secondly, they are all primes!

In the following, we will see why **97** is a prime number.

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, $$94 = 2 \times 47 = 1 \times 94$$; thus, **1, 2, 47,** and **94** are all factors of **94**, but only **2** and **47** are proper factors of **94**.

A natural number is called a **prime number** if it is greater than 1, and it doesn’t have proper factors. For example, the three primes immediately preceding **97** are **79**, **83,** and **89**.

A **composite number** is a natural number that has proper factors. As we saw, 94 has two proper factors, thus 94 is a composite number.

## Why is 97 a prime number?

Number 97 is prime because it doesn’t have proper factors. In other words, the only factors of 97 are 1 and itself. To be sure of it, we can use the following property.

**If n is a composite number, then there is a prime number less than** $$\sqrt n$$ **that divides n.**

Notice that $$97\;<\;100$$, thus $$\sqrt{97}\;<{\;\sqrt{100}\;}=\;10$$. Therefore, the prime numbers less than $$\sqrt{97}$$ are **2, 3, 5** and **7**. Moreover,

$$97 = (2\times48) + 1$$

$$97 = (3\times32) + 1$$

$$97 = (5\times19) + 2$$

$$97 = (7\times13) + 6$$

Meaning that neither of the prime numbers **2, 3, 5** nor **7** divides **97**. Then, by the property above, **97** is a prime number.

On the other hand, **a prime number of objects cannot be arranged into a rectangular grid with more than one column and more than one row.** This is another way of verifying that 97 is a prime number:

- For example, if we try to arrange 97 stars into a rectangular grid with eight rows, one of the columns will be incomplete.

The same happens if we try to arrange**97**stars into a rectangular grid with any number of rows and columns greater than one. - The only way of arranging
**97**stars into a rectangular grid, is by having a single row, or a single column. This means that**97**is a prime number!

## Which class of prime number is 97?

Number **97** is the **25 ^{th}** prime number. When we reverse its digits, we get

**79**which is also a prime number!

Moreover, **97** can be written as the sum of super-prime numbers; this is, the sum of prime numbers occupying a prime position in the list of all prime numbers: **97 = 3 + 5 + 17 + 31 + 41**; where **3, 5, 17, 31,** and **41** are the **2 ^{nd}** prime,

**3**prime,

^{rd}**7**prime, 11

^{th}^{th}prime, and

**13**prime, respectively.

^{th}Ninety-seven can be classified into several classes of primes numbers. However, as we will see next, it doesn’t belong to any of the three classes that we mention below.

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. |
No |

Mersenne prime |
It is a prime number of the form $$\operatorname{𝑝}\;=\;2_n\;-\;1$$ 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. | No |

Let’s find out why:

- Using the first three prime numbers
**2, 3,**and**5**in the primoral formula, we get

$$(2 \times 3 \times 5) − 1 = 29$$ and $$(2 \times 3 \times 5) + 1 = 31$$. Using the first four primes**2, 3, 5**, and**7**, we get

$$(2 \times 3 \times 5 \times 7) − 1 = 209$$ and $$(2 \times 3 \times 5 \times 7) + 1 = 211$$. Since the first two of the resulting numbers are less than 97, and the last two are greater than 97, then 97 doesn’t have the form of a primoral prime. - Notice that:

$$2^6\;-\;1=64-1=63$$

$$2^7\;-\;1=128-1=127$$

Therefore,**97**doesn’t have the form of a Mersenne prime. - Recall that
**47**and**53**are consecutive prime numbers. If we use them in the safe primes formula, we get**2(47) + 1 = 95**and**2(53) + 1 = 107**.

Since 97 is between**2(47) + 1 and 2(53)****+ 1**, it can’t be a safe prime.

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

You can read about 79 which is also a prime number in our next article.