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Cómo Entender el MCD y MCM: Ejercicios y Explicaciones para Niños

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Cómo Entender el MCD y MCM: Ejercicios y Explicaciones para Niños
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Ana Ruiz

@anaaaruizz_

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The document covers key concepts in number theory, focusing on divisibility, prime numbers, and common multiples. It includes explanations of the Máximo común divisor (greatest common divisor) and Mínimo común múltiplo (least common multiple), along with their properties and applications in problem-solving.

  • Explores divisibility criteria and their use in identifying multiples of specific numbers
  • Discusses prime and composite numbers, including the Goldbach conjecture
  • Provides examples and exercises to reinforce understanding of these mathematical concepts
  • Introduces the Sieve of Eratosthenes as a method for finding prime numbers

20/2/2023

8812

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 4: Divisibility and Prime Numbers

This page provides definitions and properties related to divisibility and prime numbers.

Definitions:

  • Multiples and divisors are explained in terms of integer relationships.
  • Prime numbers are defined as natural numbers greater than 1 with only two divisors: 1 and themselves.
  • Composite numbers are defined as numbers that are not prime.

Vocabulary: Prime numbers are natural numbers greater than 1 that have exactly two divisors: 1 and themselves.

Properties of divisibility are presented:

  1. If a divides b and b divides c, then a divides c.
  2. If a divides b and a divides c, then a divides b + c and b - c.
  3. If a divides b and a divides c, then a divides bc.

Highlight: The page introduces the Strong and Weak Goldbach Conjectures about expressing even and odd numbers as sums of primes.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 1: Divisibility and Multiples

This page introduces a problem involving divisibility of numbers in a specific form.

The question asks whether a number N of the form "a00a" (where 'a' is not zero) is always a multiple of 11 or 7. It then extends to another number M of the form "abba" and asks for which values of 'a' and 'b' M is a multiple of 7.

Example: The number 1001a is shown to be divisible by 11, as 1001 = 11 × 91.

Highlight: The page demonstrates how to approach problems involving divisibility of numbers with specific patterns in their digits.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 8: Sieve of Eratosthenes

This page presents the Sieve of Eratosthenes, a method for finding prime numbers.

The sieve is shown as a grid of numbers from 2 to 110, with composite numbers crossed out to reveal the prime numbers.

Definition: The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to a given limit.

Highlight: This visual representation helps students understand how prime numbers are distributed and how to identify them efficiently.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 6: Continuation of Problem 5

This page continues the solution to Problem 5 from the previous page.

The solution involves finding a number that satisfies all the given conditions:

  • n ≡ 1 (mod 2)
  • n ≡ 4 (mod 5)
  • n ≡ 5 (mod 6)
  • n ≡ 7 (mod 8)

Example: The solution shows a systematic approach to finding the number that meets all criteria, which is 1079.

Highlight: The page demonstrates how to solve complex divisibility problems using modular arithmetic concepts.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 3: Divisibility Exercises

This page presents a set of exercises related to divisibility and the greatest common divisor (GCD).

The first exercise explores divisibility criteria:

a) For a number in the form 'aba' (where a and b are non-zero), what values can a and b take for the number to be divisible by both 3 and 5?

b) What values can 'a' take for the number 20a30a to be divisible by 3?

Example: For part (a), the solution shows that a + b + a must be divisible by 3, and the number must end in 5 to be divisible by 5.

The second exercise asks to calculate possible values of GCD(12, n) where n is any natural number.

Highlight: The solution demonstrates that the possible values for GCD(12, n) are the divisors of 12: 1, 2, 3, 4, 6, and 12.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 7: More Divisibility Problems

This page presents two more problem-solving exercises related to divisibility and the least common multiple (LCM).

Problem 6: A child has a box of stamps. When grouped in 7s, 9s, or 12s, there are always 5 left over. How many stamps are there, given that the number is between 600 and 1000?

Example: The solution uses the LCM of 7, 9, and 12, which is 252, to find the answer: 761 stamps.

Problem 7: A teacher needs to arrange students in 15 rows for an exam. Placing 3 or 6 students per row leaves 2 in the last row, but 4 per row fits evenly. How many students are there, given the number is between 35 and 60?

Highlight: This problem combines divisibility rules with practical constraints to find a unique solution.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 2: Least Common Multiple (LCM)

This page defines and explains the concept of the least common multiple (LCM) and its properties.

The Mínimo común múltiplo (LCM) of two or more integers is defined as the smallest positive integer that is divisible by all the given numbers. It is calculated by multiplying all prime factors taken to the highest power in which they occur in any of the numbers.

Definition: LCM is the product of all prime factors with their highest exponents from the given numbers.

Example: LCM(180, 1050) = 2² × 3² × 5² × 7 = 6,300

Properties of LCM are discussed, including:

  1. If two numbers are multiplied or divided by the same number, their LCM is also multiplied or divided by that number.
  2. If c is a multiple of a and b, then c is also a multiple of LCM(a,b).

The relationship between the greatest common divisor (GCD) and LCM is presented:

Highlight: GCD(a,b) × LCM(a,b) = a × b

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

Page 5: Divisibility Problem Solving

This page presents two problem-solving exercises related to divisibility and the least common multiple (LCM).

Problem 4: Find all numbers between 500 and 600 that are divisible by both 3 and 8.

Example: The solution uses the LCM of 3 and 8, which is 24, to find the multiples: 504, 528, 552, 576.

Problem 5: A library has between 1000 and 1100 books. When counted in groups of 2, 5, 6, and 8, there are remainders of 1, 4, 5, and 7 respectively. How many books remain when grouped by 7?

Highlight: This problem demonstrates the use of the Chinese Remainder Theorem without explicitly naming it.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Ver

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Knowunity es la app educativa nº 1 en cinco países europeos

Knowunity fue un artículo destacado por Apple y ha ocupado sistemáticamente los primeros puestos en las listas de la tienda de aplicaciones dentro de la categoría de educación en Alemania, Italia, Polonia, Suiza y Reino Unido. Regístrate hoy en Knowunity y ayuda a millones de estudiantes de todo el mundo.

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Knowunity es la app educativa nº 1 en cinco países europeos

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Me encanta esta app [...] ¡¡¡Recomiendo Knowunity a todo el mundo!!! Pasé de un 2 a un 9 con él :D

Javi, usuario de iOS

La app es muy fácil de usar y está muy bien diseñada. Hasta ahora he encontrado todo lo que estaba buscando y he podido aprender mucho de las presentaciones.

Mari, usuario de iOS

Me encanta esta app ❤️, de hecho la uso cada vez que estudio.

Asignaturas

/

Matemáticas

/

Divisibilidad.

/

Divisibilidad de los números naturales

Cómo Entender el MCD y MCM: Ejercicios y Explicaciones para Niños

user profile picture

Ana Ruiz

@anaaaruizz_

·

1081 Seguidores

Seguir

The document covers key concepts in number theory, focusing on divisibility, prime numbers, and common multiples. It includes explanations of the Máximo común divisor (greatest common divisor) and Mínimo común múltiplo (least common multiple), along with their properties and applications in problem-solving.

  • Explores divisibility criteria and their use in identifying multiples of specific numbers
  • Discusses prime and composite numbers, including the Goldbach conjecture
  • Provides examples and exercises to reinforce understanding of these mathematical concepts
  • Introduces the Sieve of Eratosthenes as a method for finding prime numbers

20/2/2023

8812

 

1° ESO/2° ESO

 

Matemáticas

1213

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 4: Divisibility and Prime Numbers

This page provides definitions and properties related to divisibility and prime numbers.

Definitions:

  • Multiples and divisors are explained in terms of integer relationships.
  • Prime numbers are defined as natural numbers greater than 1 with only two divisors: 1 and themselves.
  • Composite numbers are defined as numbers that are not prime.

Vocabulary: Prime numbers are natural numbers greater than 1 that have exactly two divisors: 1 and themselves.

Properties of divisibility are presented:

  1. If a divides b and b divides c, then a divides c.
  2. If a divides b and a divides c, then a divides b + c and b - c.
  3. If a divides b and a divides c, then a divides bc.

Highlight: The page introduces the Strong and Weak Goldbach Conjectures about expressing even and odd numbers as sums of primes.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 1: Divisibility and Multiples

This page introduces a problem involving divisibility of numbers in a specific form.

The question asks whether a number N of the form "a00a" (where 'a' is not zero) is always a multiple of 11 or 7. It then extends to another number M of the form "abba" and asks for which values of 'a' and 'b' M is a multiple of 7.

Example: The number 1001a is shown to be divisible by 11, as 1001 = 11 × 91.

Highlight: The page demonstrates how to approach problems involving divisibility of numbers with specific patterns in their digits.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 8: Sieve of Eratosthenes

This page presents the Sieve of Eratosthenes, a method for finding prime numbers.

The sieve is shown as a grid of numbers from 2 to 110, with composite numbers crossed out to reveal the prime numbers.

Definition: The Sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to a given limit.

Highlight: This visual representation helps students understand how prime numbers are distributed and how to identify them efficiently.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 6: Continuation of Problem 5

This page continues the solution to Problem 5 from the previous page.

The solution involves finding a number that satisfies all the given conditions:

  • n ≡ 1 (mod 2)
  • n ≡ 4 (mod 5)
  • n ≡ 5 (mod 6)
  • n ≡ 7 (mod 8)

Example: The solution shows a systematic approach to finding the number that meets all criteria, which is 1079.

Highlight: The page demonstrates how to solve complex divisibility problems using modular arithmetic concepts.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 3: Divisibility Exercises

This page presents a set of exercises related to divisibility and the greatest common divisor (GCD).

The first exercise explores divisibility criteria:

a) For a number in the form 'aba' (where a and b are non-zero), what values can a and b take for the number to be divisible by both 3 and 5?

b) What values can 'a' take for the number 20a30a to be divisible by 3?

Example: For part (a), the solution shows that a + b + a must be divisible by 3, and the number must end in 5 to be divisible by 5.

The second exercise asks to calculate possible values of GCD(12, n) where n is any natural number.

Highlight: The solution demonstrates that the possible values for GCD(12, n) are the divisors of 12: 1, 2, 3, 4, 6, and 12.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 7: More Divisibility Problems

This page presents two more problem-solving exercises related to divisibility and the least common multiple (LCM).

Problem 6: A child has a box of stamps. When grouped in 7s, 9s, or 12s, there are always 5 left over. How many stamps are there, given that the number is between 600 and 1000?

Example: The solution uses the LCM of 7, 9, and 12, which is 252, to find the answer: 761 stamps.

Problem 7: A teacher needs to arrange students in 15 rows for an exam. Placing 3 or 6 students per row leaves 2 in the last row, but 4 per row fits evenly. How many students are there, given the number is between 35 and 60?

Highlight: This problem combines divisibility rules with practical constraints to find a unique solution.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 2: Least Common Multiple (LCM)

This page defines and explains the concept of the least common multiple (LCM) and its properties.

The Mínimo común múltiplo (LCM) of two or more integers is defined as the smallest positive integer that is divisible by all the given numbers. It is calculated by multiplying all prime factors taken to the highest power in which they occur in any of the numbers.

Definition: LCM is the product of all prime factors with their highest exponents from the given numbers.

Example: LCM(180, 1050) = 2² × 3² × 5² × 7 = 6,300

Properties of LCM are discussed, including:

  1. If two numbers are multiplied or divided by the same number, their LCM is also multiplied or divided by that number.
  2. If c is a multiple of a and b, then c is also a multiple of LCM(a,b).

The relationship between the greatest common divisor (GCD) and LCM is presented:

Highlight: GCD(a,b) × LCM(a,b) = a × b

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

Page 5: Divisibility Problem Solving

This page presents two problem-solving exercises related to divisibility and the least common multiple (LCM).

Problem 4: Find all numbers between 500 and 600 that are divisible by both 3 and 8.

Example: The solution uses the LCM of 3 and 8, which is 24, to find the multiples: 504, 528, 552, 576.

Problem 5: A library has between 1000 and 1100 books. When counted in groups of 2, 5, 6, and 8, there are remainders of 1, 4, 5, and 7 respectively. How many books remain when grouped by 7?

Highlight: This problem demonstrates the use of the Chinese Remainder Theorem without explicitly naming it.

12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu
12. Sea N un numero natural de la forma "à ∞0 a", con "a" distinto de O. a) ¿Es siempre N múltiplo de 11? ¿Y de 7? b) Sea M otro número natu

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Know Ejercicios de repaso 1 thumbnail

¿No encuentras lo que buscas? Explora otros temas.

Knowunity es la app educativa nº 1 en cinco países europeos

Knowunity fue un artículo destacado por Apple y ha ocupado sistemáticamente los primeros puestos en las listas de la tienda de aplicaciones dentro de la categoría de educación en Alemania, Italia, Polonia, Suiza y Reino Unido. Regístrate hoy en Knowunity y ayuda a millones de estudiantes de todo el mundo.

Ranked #1 Education App

Descargar en

Google Play

Descargar en

App Store

Knowunity es la app educativa nº 1 en cinco países europeos

4.9+

valoración media de la app

13 M

A los alumnos les encanta Knowunity

#1

en las listas de aplicaciones educativas de 12 países

950 K+

alumnos han subido contenidos escolares

¿Aún no estás convencido? Mira lo que dicen tus compañeros...

Usuario de iOS

Me encanta esta app [...] ¡¡¡Recomiendo Knowunity a todo el mundo!!! Pasé de un 2 a un 9 con él :D

Javi, usuario de iOS

La app es muy fácil de usar y está muy bien diseñada. Hasta ahora he encontrado todo lo que estaba buscando y he podido aprender mucho de las presentaciones.

Mari, usuario de iOS

Me encanta esta app ❤️, de hecho la uso cada vez que estudio.