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.
alg34-1

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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