**89 = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9**!

Isn’t that amazing? Also, 89 is a prime number with great properties as we will study 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, 98 = 2×49=14×7=1×98; thus, 1, 2, 7, 14, 49, and 98 are all factors of 98, but only 2, 7, 14, and 49 are proper factors of 98.

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

A **composite numbe**r is a natural number that has proper factors. As we saw, 98 has several proper factors, thus 98 is a composite number. Contrarily, its reverse (89) is a prime number, as we will see next.

## Why is 89 a prime number?

Number 89 is prime because it doesn’t have proper factors. In other words, the only factors of 89 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** **that divides n.**

Notice that

89 = (2×44) + 1

89 = (3×29) + 2

89 = (5×17) + 4

89 = (7×12) + 5

Meaning that neither of the prime numbers **2, 3, 5** nor **7** divides **89**. Then, by the property above, **89** 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 89 is a prime number:

- For example, if we try to arrange 89 stars into a rectangular grid with nine rows, one of the columns will be incomplete.

The same happens if we try to arrange 89 stars into a rectangular grid with any number of rows and columns greater than one. - The only way of arranging 89 stars into a rectangular grid, is by having a single row, or a single column. This means that 89 is a prime number!

## Which class of prime number is 89?

Number **89** is the **24th** prime number. It can be written, making operations with its digits, as:

Eighty-nine can be classified into several classes of primes numbers. Next, we will name three classes and see the relation of the number **89** with them.

Classes of Prime Numbers | ||

Primoral prime |
It is a prime number of the form where are the first n prime numbers. |
No |

Mersenne prime |
It is a prime number of the form 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 × 3 × 5) − 1 = 29 and (2 × 3 × 5) + 1 = 31. Using the first four primes**2, 3, 5**, and**7**, we get

(2 × 3 × 5 × 7) − 1 = 209 and (2 × 3 × 5 × 7) + 1=211. Since the first two of the resulting numbers are less than 89, and the last two are greater than 89, then 89 doesn’t have the form of a primoral prime. - Notice that:

Therefore,**89**doesn’t have the form of a Mersenne prime. However, if we use**89**as exponent, we get the**10th**Mersenne prime - Recall that
**43**and**47**are consecutive prime numbers. If we use them in the safe primes formula, we get**2(43) + 1 = 87**and**2(47 )+ 1 = 95**. Since 89 is between**2(43) + 1**and**2(47) + 1**, it can’t be a safe prime. However, using p=89 in the formula, we get**2(89) + 1 = 179**which is a prime number!

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

## Frequently Asked Questions

Yes, because its only factors are 1 and itself.

No, because it doesn’t have proper factors.

No, because it is between

**(2 × 3 × 5) ± 1** and **(2 × 3 × 5 × 7) ± 1**.

No, because it is between

No, because it is between

**2(43) + 1 = 87** and **2(47) + 1 = 95**.

### What do you think about this article? **Share your opinion with us**

**free**account to see answers!