# Is 91 a prime number?

Ninety one is a very fun number! It can be written as the sum of the first thirteen whole numbers $$91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13$$; and also as the sum of the first six squares:

$$91=1^2\;+\;2^2\;+\;3^2\;+\;4^2\;+\;5^2\;+\;6^2$$

Moreover, **91** is closely related to prime numbers, as we will see 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, $$9 = 3 \times 3 = 1 \times 9$$; thus, **1, 3,** and** 9** are all factors of **9**, but only **3** is a proper factor of **9**.

A natural number is called a **prime number** if it is greater than **1**, and it doesn’t have proper factors. For example, the prime numbers closer to **91** are: **89** and **97**.

A **composite number** is a natural number that has proper factors. For example, **9** is a composite number because we just saw it has one proper factor. As we will see next, **91** is also a composite number.

## Why is 91 not a prime number?

There are several ways of showing that **91** is not a prime number. For example, by finding a proper factor for it. We know that a proper factor for **91** is a divisor which is between **2** and **90**. Thus, we could try with every number in the list **2, 3, 4, …, 89, 90** to see if one of them divides **91**. But this is a long list…! Fortunately, we don’t need to try with each of them. Instead, 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 $$91\;<\;100$$ , thus $$\sqrt{91}\;<\;\sqrt{100}\;=\;10$$. Therefore, the prime numbers less than $$\sqrt{91}$$ are** 2, 3, 5,** and **7**. Hence, if **91** is a composite number, one of these four primes must divide **91**.

Since **91** is not an even number, **2** doesn’t divide **91**. Thus, we verify if **3** divides **91**, and we get that it does not:

$$91 = (3 \times 30) + 1$$.

Then, we verify if **5** divides **91**, and again we get that it does not:

$$91 = (5 \times 18) + 1$$.

We continue with **7** (our last hope!), and indeed we see that **7** divides **91**:

$$91 = 7 \times 13$$.

Therefore, **7** and **13** are proper factors of **91**. This means that **91** is a composite number, it isn’t prime!

However, although 91 is not a prime number, it is the product of exactly two primes: **7** and **13**.

Another way of understanding why 91 is not prime, is recalling that **a prime number of objects cannot be arranged into a rectangular grid with more than one column and more than one row.** As we see below, 91 stars can certainly be arranged into a rectangular grid with 7 rows and 13 columns. This means that 91 is not a prime number!

$$91 = 7 \times 13$$

Notice that the 91 stars can also be arranged into a rectangular grid with 13 rows and 7 columns. Can you draw such a grid?

## Occurrences of 91 among the prime numbers

Since **91** is the product of exactly two prime numbers: $$91 = 7 \times 13$$, it is what is called a semi-prime number.

Sometimes, **91** can be confused with a prime number because it behaves as *twin prime*. A twin prime is a prime number that is 2 less or 2 more than another prime number. Notice that $$91 = 89 + 2$$, and **89** is a prime number; but since **91** is not prime, it isn’t a twin prime.

We list below the first few occurrences of **91** among the prime numbers so that you can have them at hand when deciding if some numbers related to **91** are primes.

First few prime numbers where 91 occurs | 191, 491, 691, 911, 919, 991 |

**Example**: Which of the numbers 1**91** and 2**91** is prime?

We first notice that 191 is in the table. Therefore, we know 191 is a prime number.

However, 291 is not in the table. Thus, 291 should be a composite number. Since **291 < 324**, we have that $$\sqrt{291}\;<{\;\sqrt{324}}\;=\;18$$. Thus, one of the primes **2, 3, 5, 7, 11, 13,** or **17** must be a factor of **291**. It is easy to see that **3** is a factor of **291**, because $$291 = 3 \times 97$$. Therefore, **291** is a composite number!

Do you know that 89 is a prime number?