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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.
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?
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.
Example: Which of the numbers 191 and 291 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?
No, because it has proper factors.
Yes, because 7 and 13 are proper factors of 91.
Number 91 has four factors: 1, 7, 13 and 91.
Number 91 has two prime factors: 7 and 13.
No, because 91 is not a prime number.
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