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What do the numbers 900007, 90007, 9007, 907 and 97 have all in common? Firstly, each of themis 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.
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, $$94 = 2 \times 47 = 1 \times 94$$; 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.
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,
$$97 = (2\times48) + 1$$$$97 = (3\times32) + 1$$$$97 = (5\times19) + 2$$$$97 = (7\times13) + 6$$
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:
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.
Let’s find out why:
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.
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 $$2^6\;–\;1=63$$ and $$2^7\;-\;1=127$$.
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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