Prime and composite numbers

Prime numbers

If we try to find divisors of number $$13$$ we will see that it only has itself and the unit as divisors.

$$$13 \div 13=1$$$

$$$13 \div 1=13$$$

Therefore, it will not be a multiple of any number, apart from the $$1$$ and $$13$$.

$$$13 \times 1=13$$$

It is said that it is a prime number. The prime numbers are, therefore, those that can only be divided by themselves and the unit.

Here is a list with the first 25 prime numbers:

$$$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89\mbox{ i }97$$$

To find out if a number is prime it is necessary to try to divide it by the prime numbers less than itself. If it is really a prime, none of these divisions will be exact. The moment the quotient is equal or less than the divisor, we can conclude that this is a prime number.

If we want to verify if number $$157$$ is prime, we have to do the following divisions:

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In this latter division we have already obtained a quotient ($$12$$) less than the divisor ($$13$$), therefore it is not necessary to keep on dividing any more: it is confirmed that number $$157$$ is prime.

With the number $$239$$ more divisions are needed to reach the same conclusion:

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In the latter division, the quotient ($$14$$) is less than the divisor ($$17$$), so we can already confirm that $$239$$ is really a prime number.

Composite numbers

A composite number is a number that is not prime, that is, it has more than two divisors: itself, the unit, and other numbers.