## The definition of a Prime number

**Definition: **a Natural number is called simpleif it has only two positive divisors — the number itself and 1.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, ... — the Prime numbers.

Prime numbers are infinitely many.

## The definition of composite numbers

**Definition: **a Natural number is called compositeif it has more than two positive divisors.

6, 15, 130, ... are composite numbers.

1 is neither Prime number nor composite.

## Properties of Prime factors of natural numbers

- Any natural number (greater than unity) or is divided by this Prime number , or is
**coprime**with it. - If the product of several factors is divisible by a Prime number , then at least one of the factors is divisible by .
- The smallest Prime divisor of a composite number not to .

## The main theorem of the theory of divisibility

**Theorem:** Any natural number greater than unity, can be decomposed into the product of primes, and this decomposition is unique up the order of the multiplicands.

where are all Prime numbers.

## The theorem on division with Stacey

**Theorem:** For any pair of numbers and there is, and only one pair of integers and such that , where ( is the incomplete quotient of at , — modulo on ).