What are prime numbers?

Prime numbers are those that have only two divisors: one and the number itself. They are part of the set of natural numbers.

For example, 2 is a prime number as it is only divisible by one and itself.

When a number has more than two divisors they are called composite numbers and can be written as a product of prime numbers.

For example, 6 is not a prime number, it is a composite number as it has more than two divisors (1, 2 and 3) and is written as the product of two prime numbers 2 x 3 = 6.

Some considerations about prime numbers:

• The number 1 is not a prime number as it is only divisible by itself;
• The number 2 is the smallest prime number and also the only one that is even;
• The number 5 is the only prime number ending in 5;
• The other prime numbers are odd and end with the digits 1, 3, 7 and 9.

How do you know if a number is prime?

One way to find a prime number is to use the Sieve of Eratosthenes.

1. Create a table and write the numbers in a range, for example from 1 to 100.
2. The number 1 can be eliminated as it is not a prime number.
3. Mark all prime numbers less than 10 (2, 3, 5 and 7) with different colors.
4. Eliminate multiples of these numbers by marking them with their respective colors.
5. The remaining numbers in the table, which have not been checked, are the prime numbers.

From the table we can see that there are 25 prime numbers between 1 and 100. Are they:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

Another way to recognize a prime number is to perform divisions with the investigated number. To make the process easier, see some divisibility criteria.

Divisibility by 2: every number whose unit digit is even is divisible by 2;

Divisibility by 3: a number is divisible by 3 if the sum of its digits is a number divisible by 3;

Divisibility by 5: a number will be divisible by 5 when the unit digit is equal to 0 or 5.

If the number is not divisible by 2, 3 and 5 we continue the divisions with the next prime numbers less than the number until:

• If it is an exact division (rest equals zero) then the number is not prime.
• If it is an inexact division (non-zero remainder) and the quotient is smaller than the divider, then the number is prime.
• If it is an inexact division (non-zero remainder) and the quotient is equal to the divisor, then the number is prime.

Solved example: check if the number 113 is prime.

About number 113, we have:

• It does not have the last even digit and, therefore, is not divisible by 2;
• The sum of its digits (1+1+3 = 5) is not a number divisible by 3;
• It doesn't end in 0 or 5, so it's not divisible by 5.

As we have seen, 113 is not divisible by 2, 3 and 5. Now, it remains to be seen whether it is divisible by prime numbers smaller than it using the division operation.

Division by prime number 7:

Division by prime number 11:

Note that we have arrived at an inexact division whose quotient is less than the divisor. This proves that the number 113 is prime.

Prime numbers from 1 to 1000

Check out the 168 prime numbers between 1 and 1000.

Prime numbers from 1 to 10:
2, 3, 5, 7
Prime numbers from 10 to 100:
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Prime numbers from 100 to 200:
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
Prime numbers from 200 to 300:
211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293
Prime numbers from 300 to 400:
307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397
Prime numbers from 400 to 500:
401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499
Prime numbers from 500 to 600:
503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599
Prime numbers from 600 to 700:
601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691
Prime numbers from 700 to 800:
701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797
Prime numbers from 800 to 900:
809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887
Prime numbers from 900 to 1000:
907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

Also read about:

• dividers
• Multiples and Dividers
• What are prime numbers?