Is 71 a prime number?

Did you know that about 71% of Earth is in fact water? That is why, from space, it looks like a beautiful blue planet.

Number 71 has many more interesting properties For example, if we reverse its digits, we get 17 which is a prime number; and, as we will see next, so is 71.

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, $$74  =  2 \times 37 = 1 \times 74$$; thus, 1, 2, 37 and 74 are all factors of 74, but only 2 and 37 are proper factors of 74.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the ten prime numbers immediately below 71 are: 29, 31, 37, 41, 43, 47, 53, 59, 61 and 67.

A composite number is a natural number that has proper factors. As we saw, 74 has two proper factors, thus it is a composite number.

Why is 71 a prime number?

Number 71 is prime because it doesn’t have proper factors. In other words, the only factors of 71 are 1 and itself. To be sure of it, we can use the following property.

If 𝒏 is a natural number, and neither of the prime numbers less than $$\sqrt n$$ divides 𝒏, then 𝒏 is a prime number.

Notice that 71<81, thus $$\sqrt{71}\;<\;\sqrt{81}\;=\;9$$. Therefore, the prime numbers less than $$\sqrt{71}$$ are 2, 3, 5 and 7. Moreover,

$$71 = (2\times35) + 1$$
$$71 = (3\times23) + 2$$
$$71 = (5\times14) + 1$$
$$71 = (7\times10) + 1$$

Meaning that neither of the prime numbers 2, 3, 5 nor 7 divides 71. Then, by the property above, 71 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 71 is a prime number:

• For example, if we try to arrange 71 stars into a rectangular grid with seven rows, one of the columns will be incomplete.
  
  The same happens if we try to arrange 71 stars into a rectangular grid with any number of rows and columns greater than one.
• The only way of arranging 71 stars into a rectangular grid, is by having a single row, or a single column. This means that 71 is a prime number!

Which class of prime number is 71?

Seventy-one is the 20th prime number. Since 71 = 73 – 2, and 73 is also a prime number, then 71 is a twin prime: this is, a prime number that is 2 less or 2 more than another prime number.

Amazingly, 71 divides the sum of all prime numbers less than or equal to 71:

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71
= 639 = 9 × 71.

Seventy-one 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 $$p\;=\;2^n\;\;-\;1$$ 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 $$(2\times3\times5)−1=29$$ and $$(2\times3\times5)+1=31$$. Using the first four primes 2, 3, 5, and 7, we get $$(2\times3\times5\times7)−1=209$$ and $$(2\times3\times5\times7)+1=211$$. Since the first two of the resulting numbers are less than 71, and the last two are greater than 71, then 71 doesn’t have the form of a primoral prime.
• Notice that:
  $$2^6\;-\;1\;=\;64\;-\;1\;=\;63$$
  $$2^7\;-\;1\;=\;128\;-\;1\;=\;127$$
  Therefore, 71 doesn’t have the form of a Mersenne prime.
• Recall that 31 and 37 are consecutive prime numbers. Moreover, if we use them in the safe primes formula, we get 2(31) + 1 = 63 and 2(37) + 1 = 75. Since 71 is between 2(31) + 1 and 2(37) + 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.

In the next article you can read about 17 which is a prime number.