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 ∈ N*o 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

o*n; (ab) c = a (*

bc) -distribution law of multiplication with respect to addi

t*ion; (a + b) c = a*

s + bc – zero is a neutral element for additi

o*n; а + 0 = 0 + а =*

*а а • 0 = 0 • а = 0*

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

For given *a, b *∈ N, the differen*ce 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 give*n a,* b ∈ N. the quotie*nt a: *b is not always determined, ie. is not a natural number.

##### 2.1. DISTRIBUTION

For any given natural numbers a a*n*d b*,* a*>* b,* *there are uniquely defined numbers k, *r ∈* No such that *a = kb + r*, 0 = <*r* <b,

where k i*s* 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 numbe*r* b, if there exists a natural num*b*er k, so

what:

** a = kb**.

The rule of divisibility is symbolically written:

** b | a <=> а = k**b,

*k ∈ N.*

– Any natural number n tha*t* is divisible by the number* 2* is called* an even num*ber 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 t*h*at has exactly two divisors 1 and n i*s* 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 i

**umber (divisibility sign by 2).**

*s an even n*2. **5 | a <=> **a₀

**i**s 0 o

**r**5 (divisibility sign by 5).

3. **10 | a <=**

*a₀*

**>****i**s 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 <=> **

*₁а₀ (Divisibility sign with 4).*

**4 |**а##### 2.5. LOWEST COMMON DENOMINATOR

The smallest common denomi* nator (N*SS) of two or more natural numbers is calle

**s.**

*d the smallest number that contains each of the given number*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.*