57 Is A Prime Number

Fifty-7 is closely related to prime numbers. For case, 57's digits are both prime number numbers: five and 7. As well, 57 tin can be written using the primes numbers ii and v:

$$57\;=\;five^2\;+\;2^v\;$$

Yet, 57 itself is not a prime number, every bit we will study next.

Nosotros will start recalling some definitions that we take widely discussed in our Prime Numbers commodity. If you still feel unfamiliar with these notions, we invite you to first read that article and come dorsum here later.

A cistron of a natural number is a positive divisor of the number. A proper factor of a natural number is a gene that is dissimilar from 1 and from the number itself.

For example, 75 = 5 × 15 = three × 25 = one × 75; thus, ane, 3, 5, 15, 25 and 75 are all factors of 75, but only 3, 5, 15 and 25 are proper factors of 75.

A natural number is called a prime number if it is greater than 1, and it doesn't have proper factors. For instance, the prime number numbers closer to 57 are 53 and 59.

A composite number is a natural number that has proper factors. For
case, 75 is a composite number because, as we just saw, it has several
proper factors. Equally we will encounter next, 57 is also a blended number.

Why is 57 not a prime number?

There are several ways of showing that 57 is not a prime number. For instance, we tin can use the following property, that we discussed in our Prime number Numbers article.

Every prime number is of the form 6k+1 or 6k+5: Therefore, if a number is non of the grade 6k+1 or 6k+five, information technology tin can't be prime. In other words, if the residuum when a number is divided by half dozen is dissimilar from i or 5, then the number is non prime number.

Notice that 57 = 6(9) + 3. Thus, the residue of dividing 57 by 6 is 3, which is different from 1 and 5. Therefore, 57 is not prime, as the property in a higher place indicates

This means that 57 is a composite number. Thus, 57 has proper factors. In club to notice them, we will use the next property, which we usually phone call the "Square Root Rule".

If 𝒏 is a blended number, and then in that location is a prime number number less than $$\sqrt n$$ that divides 𝒏.

Notice that 57<64, thus $$\sqrt{57}<\sqrt{64}=8$$. Therefore, the prime number numbers less than $$\sqrt{57}$$ are 2, three, 5, and 7; and one of them must split 57. Since 57 is non an even number, 2 doesn't divide 57. Thus, we verify if 3 divides 57, and we become that it does:

57 = three × xix.

Therefore, 3 and nineteen are proper factors of 57. Moreover, since 3 and nineteen are both prime numbers, these are the only proper factors of 57.

Summing up, we have that:

The factors of 57 are one, 3, xix, and 57, because 57 = 3 × nineteen = 1 × 57.
The proper factors of 57 are iii and 19.
57 is a composite number.
57 is non a prime.

Another way of understanding why 57 is not prime, is recalling that a prime of objects cannot be arranged into a rectangular grid with more than than one cavalcade and more than 1 row.

As we see below, 57 stars can certainly exist arranged into a rectangular grid with 19 columns and 3 rows. This ways that 57 is not prime number!

Occurrences of 57 amongst the prime numbers

Although 57 is not a prime, it is a semiprime number; that is, a natural number that is the production of exactly ii prime number numbers: 57 = iii × 19, three and 19 are prime number numbers.

On the other hand, in that location are many occurrences of 57 among the prime number numbers. We list below the first few of them. This way, yous tin have them at hand when deciding if some numbers related to 57 are primes.

First few prime numbers where 57 occurs

157, two57, four57, 557, 571

Case: Which of the numbers 357 and 457 is prime?

We first notice that 457 is in the tabular array. Therefore, we know 457 is a prime number number.

We too observe that 357 is non in the table. Thus, 357 should be a composite number. Moreover, since 357<361, we take that

$$\sqrt{357}<\sqrt{361}=19$$

Thus, by the Foursquare Root Rule, one of the primes two, 3, 5, seven, xi, 13, or 17 must be a gene of 357. Is easy to meet that 3 is a factor of 357 considering 357 = 3 × 119. Therefore, 357 is a blended number!

Do you think 59 is a prime number number or not?

Frequently Asked Questions

: No, considering information technology has proper factors
: Yes, considering three and xix are proper factors of 57.
: Number 57 has four factors: ane, 3, 19 and 57.
: Number 57 has 2 prime factors: 3 and nineteen.
: Aye, considering 57=3×19. So, 57 divided by 3 is equal to nineteen.