Fifty-seven is closely related to prime numbers. For example, 57’s digits are both prime numbers: 5 and 7. Also, 57 can be written using the primes numbers 2 and 5:
However, 57 itself is not a prime number, as we will study 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, 75 = 5 × 15 = 3 × 25 = 1 × 75; thus, 1, 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 example, the prime numbers closer to 57 are 53 and 59.
A composite number is a natural number that has proper factors. For
example, 75 is a composite number because, as we just saw, it has several
proper factors. As we will see next, 57 is also a composite number.
Why is 57 not a prime number?
There are several ways of showing that 57 is not a prime number. For example, we can use the following property, that we discussed in our Prime Numbers article.
Every prime number is of the form 6k+1 or 6k+5: Therefore, if a number is not of the form 6k+1 or 6k+5, it can’t be prime. In other words, if the remainder when a number is divided by 6 is different from 1 or 5, then the number is not prime.
Notice that 57 = 6(9) + 3. Thus, the remainder of dividing 57 by 6 is 3, which is different from 1 and 5. Therefore, 57 is not prime, as the property above indicates
This means that 57 is a composite number. Thus, 57 has proper factors. In order to find them, we will use the next property, which we usually call the “Square Root Rule”.
If 𝒏 is a composite number, then there is a prime number less than that divides 𝒏.
Notice that 57<64, thus . Therefore, the prime numbers less than are 2, 3, 5, and 7; and one of them must divide 57. Since 57 is not an even number, 2 doesn’t divide 57. Thus, we verify if 3 divides 57, and we get that it does:
57 = 3 × 19.
Therefore, 3 and 19 are proper factors of 57. Moreover, since 3 and 19 are both prime numbers, these are the only proper factors of 57.
Summing up, we have that:
- The factors of 57 are 1, 3, 19, and 57, because 57 = 3 × 19 = 1 × 57.
- The proper factors of 57 are 3 and 19.
- 57 is a composite number.
- 57 is not a prime number.
Another way of understanding why 57 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, 57 stars can certainly be arranged into a rectangular grid with 19 columns and 3 rows. This means that 57 is not prime!
Occurrences of 57 among the prime numbers
Although 57 is not a prime number, it is a semiprime number; that is, a natural number that is the product of exactly two prime numbers: 57 = 3 × 19, 3 and 19 are prime numbers.
On the other hand, there are many occurrences of 57 among the prime numbers. We list below the first few of them. This way, you can have them at hand when deciding if some numbers related to 57 are primes.
| First few prime numbers where 57 occurs | 157, 257, 457, 557, 571 |
Example: Which of the numbers 357 and 457 is prime?
We first notice that 457 is in the table. Therefore, we know 457 is a prime number.
We also notice that 357 is not in the table. Thus, 357 should be a composite number. Moreover, since 357<361, we have that
Thus, by the Square Root Rule, one of the primes 2, 3, 5, 7, 11, 13, or 17 must be a factor of 357. Is easy to see that 3 is a factor of 357 because 357 = 3 × 119. Therefore, 357 is a composite number!
Do you think 59 is a prime number or not?
