Własności liczb naturalnych

Learning objectives

• You will consolidate your knowledge of natural numbers.

Learning effect

You use your knowledge of natural numbers.

• Ways of writing and reading numbers. Positional decimal systemdecimal systemdecimal system. Large natural numbers.

• Properties of operations in the set of natural numbers.

• Prime numbers and composite numbers. Features of divisibility of numbers.

Prepare the information on the following topics.

1. Ways of writing and reading numbers. Positional decimal system. Large natural numbers.

2. Properties of operations in the set of natural numbers.

3. Prime numbers and composite numbers. Features of divisibility of numbers.

4. Prime factorization. The Greatest Common Divisor (GCD), the Least Common Multiple (LCM).

Check if the information you prepared contains the information provided below.

1. Ways of writing and reading numbers. Positional decimal system. Large natural numbers.

We use Arabic numerals to create numbers nowadays: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The number system we use at present is called the positional decimal systemdecimal systemdecimal system.

The position of a digit in a number can be determined using the scheme below:

• 1 000 000 - million

• 1 000 000 000 - billion

• 1 000 000 000 000 - trillion

• 1 000 000 000 000 000 – quadrillion

• 1 000 000 000 000 000 000 – quintillion

• 1 000 000 000 000 000 000 000 – sextillion

2. Properties of operations in the set of natural numbers.

Natural numbers can be added, subtracted, divided and squared or cubed.

The order of performing operations:

1. operations in brackets,

2. exponentiation,

3. multiplication or division - in the order of writing,

4. addition or subtraction - in the order of writing.

Properties of operations:

1. addition is associative and commutative,

2. multiplication is associative and commutative,

3. we do not divide by zero.

3. Prime numbers and composite numbers. Features of divisibility of numbers.

The prime numberprime numberprime number is a natural number greater than 1, whose only divisors are 1 and itself.

A composite numbercomposite numbercomposite number is a natural number greater than 1 that has more than two divisors.

Numbers 0 and 1 are neither prime numbers nor composite numbers.

Features of divisibility of natural numbers. The natural number is divisible:

• by 2 if its unity digit is 0, 2, 4, 6 or 8,

• by 3 if the sum of its digits is a number divisible by 3,

• by 4 if its last two digits form a number divisible by 4,

• by 5 if its unity digit is 0 or 5,

• by 9, if the sum of its digits is a number divisible by 9,

• by 25 if its last two digits form a number divisible by 25.

4. Prime factorization. The Greatest Common Divisor (GCD), the Least Common Multiple (LCM).

Prime factorization is the representation of this number in the form of a product of prime numbers.

GCD (a, b) - the greatest common divisor of natural numbers a and bgreatest common divisor of natural numbers a and bgreatest common divisor of natural numbers a and b, is the largest natural positive number, which is also the divisor of the number a and the divisor of the number b.

LCM (a, b) - the least common multiple of the numbers a and b, the smallest positive natural number that is divisible by the number of a and the number of b.

Tasks for Topic 1

Tasks for Topic 2

Tasks for Topic 3

Task 6

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nagranie abstraktu

nagranie abstraktu

Nagranie dostępne na portalu epodreczniki.pl

nagranie abstraktu

Open the applet. Using the features of divisibility, put the numbers in the right places.

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Geogebra aplet - podzielność liczb naturalnych. Galeria z opisami alternatywnymi poniżej.

Geogebra aplet - podzielność liczb naturalnych. Galeria z opisami alternatywnymi poniżej.

Zasób interaktywny dostępny pod adresem https://zpe.gov.pl/a/D1DmRTf4H

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Rysunek przedstawia dwa zbiory – zbiór liczb dzielących się przez 2 oraz zbiór liczb dzielących się przez 9. Zbiory są niezapełnione. Pod nimi wypisane są liczby, które należy wpisać w zbiorach w odpowiednim miejscu. Te liczby to: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.

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Rysunek przedstawia dwa zbiory – zbiór liczb dzielących się przez 2 oraz zbiór liczb dzielących się przez 9. W pierwszym zbiorze nie znajduje się żadna liczba. W drugim zbiorze znajdują się liczby 9, 27, 45, 63, 81. W części wspólnej zbiorów wpisane są liczby 18, 36, 54, 72 i 90.

Tasks for Topic 4

Remember:

• We use Arabic numerals to create numbers nowadays: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

• The number system we use at present is called the positional decimal systemdecimal systemdecimal system.

• Natural numbers can be added, subtracted, divided and squared or cubed.

• The order of performing operations:

  1. operations in brackets,

  2. exponentiation,

  3. multiplication or division - in the order of writing,

  4. addition or subtraction - in the order of writing.

• Addition and Multiplication are  Associative and Commutative.

• We do not divide by zero.

• The prime numberprime numberprime number is a natural number greater than 1, whose only divisors are 1 and itself.

• A composite numbercomposite numbercomposite number is a natural number greater than 1 that has more than two divisors.

• GCD (a, b) - the greatest common divisor of natural numbers a and bgreatest common divisor of natural numbers a and bgreatest common divisor of natural numbers a and b, is the largest natural positive number, which is also the divisor of the number a and the divisor of the number b.

• LCM (a, b) - the least common multiple of the numbers a and b, the smallest positive natural number that is divisible by the number of a and the number of b.

Exercises

Glossary

Keywords

composite numbercomposite numbercomposite number - a natural number greater than 1 that has more than two divisors

greatest common divisor of natural numbers a and bgreatest common divisor of natural numbers a and bgreatest common divisor of natural numbers a and b - the largest natural positive number, which is also the divisor of the number a and the divisor of the number b, order of operations

prime numberprime numberprime number - a natural number greater than 1, which has only two divisors: 1 and itself

least common multiple of numbers a and bleast common multiple of numbers a and bleast common multiple of numbers a and b - the smallest positive natural number that is divisible by the number a and the number b