Not only **101** but also its three powers

Moreover, **101** is a prime number, 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, **10 = 2 × 5 = 1 × 10**; thus, **1, 2, 5** and **10** are all factors of **10**, but only **2** and **5** are proper factors of **10**.

A natural number is called **prime number** if it is greater than **1**, and it doesn’t have proper factors. For example, the three primes immediately preceding **101** are **83**, **89** and **97**.

A **composite number** is a natural number that has proper factors. As we saw, **10** has two proper factors, thus **10** is a composite number.

## Why is 101 a prime number?

Number **101** is prime because it doesn’t have proper factors. In other words, the only factors of **101** 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 **101 < 121**, thus **2, 3, 5** and **7**. Moreover,

101=(2×50)+1

101=(3×33)+2

101=(5×20)+1

101=(7×14)+3

Meaning that neither of the prime numbers **2, 3, 5** nor **7** divides **101**. Then, by the property above, **101** 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 **101** is a prime number:

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

The same happens if we try to arrange 101 stars into a rectangular grid with any number of rows and columns greater than one. - The only way of arranging 101 stars into a rectangular grid, is having a single row, or a single column. This means that 101 is a prime number!

## Which class of prime number is 101?

Number **101** is the **26th** prime number. Since **101 = 103 – 2**, and **103** is also a prime number, then 101 is a *twin prime*: this is, a prime number that is **2** less or **2** more than another prime number.

Since the reverse of **101** is itself, **101** is a palindromic prime. Moreover, **101** can be written as the sum of five consecutive prime numbers:

**101 = 13 + 17 + 19 + 23 + 29**

One hundred and one (**101**) can be classified into several classes of prime 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 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**101**, and the last two are greater than**101**, then**101**doesn’t have the form of a primoral prime. - Notice that:

Therefore,**101**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 101 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 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(47)+1=95 and 2(53)+1=107, where 47 and 53 are consecutive prime numbers.

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