Podzielność liczb naturalnych

Learning objectives

You will develop accounting competency in the field of performing operations on natural numbers.

You will improve and systematize your knowledge of divisibility in a set of natural numbers.

Learning effect

• You do operations in a set of natural numbers.

• You use the theory of divisibility to solve arithmetical problems.

Divisibility rules of numbers.

Division:

• by 2 - the number is divisible by 2 if the digit of unity is 0, 2, 4, 6, 8,

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

• by 4 - the number is divisible by 4 if it contains two last digits divisible by four or whose two last digits are zeros,

• by 5 - the number is divisible by 5 if the last digit of this number is 0 or 5,

• by 6 - the number is divisible by 6 if it is divisible simultaneously by 2 and by 3,

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

• by 10 - the number is divisible by 10 if the last digit of this number is 0,

• by 25 - the number is divisible by 25 if its last two digits are divisible by 25 or the last two digits are zeros,

• by 100 - the number is divisible by 100 if its last two digits are zeros.

Recall the most important divisibility rules of numbers. Think how, by not doing the division, one can check whether the number is divisible by 12, by 15 or by 18.

Make your hypotheses and then prove it by solving the following problems.

Task 2

Observe the calculation of LCM and GCD of two numbers. Then do analogous calculation of LCM and GCD of numbers 324, 243, 289.

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Pokaz multimedialny przedstawia sposób wyznaczania NWW i NWD dwóch liczb. Instrukcja obsługi z poziomu klawiatury: 1. Uruchomienie aplikacji - ENTER, 2. Na każdym ze slajdów czytany jest automatycznie tekst alternatywny po polsku, 3. Przy pierwszym uruchomieniu na pierwszym slajdzie, czytanie tekstu po angielsku - TAB, 4. Przejście między slajdami: do następnego slajdu - TAB, do poprzedniego slajdu - TAB + SHIFT, 5. Przejście do czytania napisu po angielsku - strzałka w górę + strzałka w dół (czyta tekst po angielsku widoczny na slajdzie).

Pokaz multimedialny przedstawia sposób wyznaczania NWW i NWD dwóch liczb. Instrukcja obsługi z poziomu klawiatury: 1. Uruchomienie aplikacji - ENTER, 2. Na każdym ze slajdów czytany jest automatycznie tekst alternatywny po polsku, 3. Przy pierwszym uruchomieniu na pierwszym slajdzie, czytanie tekstu po angielsku - TAB, 4. Przejście między slajdami: do następnego slajdu - TAB, do poprzedniego slajdu - TAB + SHIFT, 5. Przejście do czytania napisu po angielsku - strzałka w górę + strzałka w dół (czyta tekst po angielsku widoczny na slajdzie).

Zasób interaktywny dostępny pod adresem https://zpe.gov.pl/a/DMUnuijP8

Think what the relationship between the product of two numbers, their least common multiplethe Least Common Multiple (LCM)least common multiple and the greatest common divisorthe Greatest Common Divisor (GCD)greatest common divisor is.

Compare your conclusions with the formula:

where:
a and b - are positive natural numbersnatural numbersnatural numbers.

Perform revision exercises.

Remember:

• To check whether a given number is divisible by another, we use methods called divisibility rules.

• The prime factorization can be used to calculate their least common multiplethe Least Common Multiple (LCM)least common multiple and their greatest common divisorthe Greatest Common Divisor (GCD)greatest common divisor.

• The relationship between the product of two numbers, their least common multiplethe Least Common Multiple (LCM)least common multiple of the greatest common divisorthe Greatest Common Divisor (GCD)greatest common divisor is expressed by the formula:

  $LCM(a,b)=\frac{a\cdot b}{GCD(a,b)}$

  where:
  a and b - are positive natural numbersnatural numbersnatural numbers.

Exercises

Glossary

Keywords

composite numberscomposite numberscomposite numbers - liczby naturalne, które mają więcej niż dwa dzielniki

divisorsdivisorsdivisors

natural numbersnatural numbersnatural numbers

prime numbersprime numbersprime numbers - liczby naturalne, które mają tylko dwa dzielniki jedność i samą siebie

relatively prime numbers (coprime numbers)relatively prime numbers (coprime numbers)relatively prime numbers (coprime numbers)