# Natural numbers

The set of natural numbers M = {1, 2, 3, …}.

The number 1 is the smallest natural number, and the largest natural number does not exist.

The set whose elements are all natural numbers and the number zero is denoted as N0 = {0, 1,2,3, …}.

#### 1. ADDITION AND MULTIPLICATION OF NATURAL NUMBERS

The sum and product of any two natural numbers is a natural number, ie.

(∀a, b ∈ N) (a + b, ab ∈ N)

For any a, b, c ∈ No the following equations are true:
– commutative law of addition; a +
b = b + a – a
ssociative law of addition; (a + b)
+ c = a + (b + c) – the uni
t is a neutral element for multiplication; a •
1 = 1 • a = a – co
mmutative law of multiplication
a • b = b • a
– associative law of multiplicati
on; (ab) c = a (
bc) -distribution law of multiplication with respect to addi
tion; (a + b) c = a
s + bc – zero is a neutral element for additi
on; а + 0 = 0 + а =
а а • 0 = 0 • а = 0

#### 2. SUBTRACTION AND DIVISION OF NATURAL NUMBERS

For given a, b ∈ N, the difference a – b is not always determined, ie. does not exist in N.
If a> b, then a – b ∈ N. (In Nₒ, the condition is a> = b.) For a given a, b ∈ N. the quotient a: b is not always determined, ie. is not a natural number.

##### 2.1. DISTRIBUTION

For any given natural numbers a and b, a> b, there are uniquely defined numbers k, r ∈ No such that a = kb + r, 0 = <r <b,
where k is a quotient and r is the remainder of dividing a by b.

##### 2.2. DELIVERY RELATIONSHIP

The natural number a is divisible by the natural number b, if there exists a natural number k, so
what:
a = kb.

The rule of divisibility is symbolically written:
b | a <=> а = kb, k ∈ N.

– Any natural number n that is divisible by the number 2 is called an even number and is denoted by
n = 2k, for some k ∈ N.

– If n is not divisible by 2, then it is called an odd number and is denoted by n = 2k + 1, k ∈ N.

##### 2.3. SIMPLE AND COMPLEX NUMBERS

Any natural number n that has exactly two divisors 1 and n is called a prime number.
Natural numbers that have more than two divisors are called complex numbers.
The number 1 is neither a simple nor a complex number.

##### 2.4. RECOGNITIONS OF DELIVERY

Any Natural number a can be represented by the degrees based on the number 10, ie. where a₀ is the number of units, a₁ – tens, etc.

– Signs of divisibility:

1. 2 | a <=> a₀ is an even number (divisibility sign by 2).

2. 5 | a <=> a₀ is 0 or 5 (divisibility sign by 5).

3. 10 | a <=> a₀ is 0 (divisibility sign by 10).

4. 3 | а <=> 3 | (аn + аn-1 + … + а, a0) (sign of divisibility by 3).

5. 9 | а <=> 9 | (аn + аn-1 + … + а, a0) (sign of divisibility by 9).

6. 4 | a <=> 4 | а₁а₀ (Divisibility sign with 4).

##### 2.5. LOWEST COMMON DENOMINATOR

The smallest common denominator (NSS) of two or more natural numbers is called the smallest number that contains each of the given numbers.

Example. NHS (60,45,48):

60 = 2² • 3 • 5
45 = 3² • 5
48 = 2⁴ • 3
So, NHS (60,45,48) = 2⁴ ∙ 3² ∙ 5 = 720

##### 2.6. GREATEST COMMON DIVISOR

The largest common divisor (NPD) of two or more natural numbers is called the largest natural number that divides those numbers.

Example NZD (60,90,48)
: 60 = 2² • 3 •
5 90 = 2 • 3²
• 5 48 = 2
⁴ • 3 So
, NZD (60,90,48) = 2 • 3 = 6.

If the NZD of two numbers is 1, then those numbers are said to be mutually prime.

###### – Natural numbers –
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