Is 97 a prime number?

6 min read
Is 97 a prime number?

is 97 composite or primeWhat 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.

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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 = 2×47=1×94; 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.

Is 97 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 = (2×48) + 1
97 = (3×32) + 1
97 = (5×19) + 2
97 = (7×13) + 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.
    What is the prime factorization of 97The 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 25th 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 2nd prime, 3rd prime, 7th prime, 11th prime, and 13th prime, respectively.

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 × 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 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.

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