Números PRIMOS e COMPOSTOS

Summary

En este video, el canal de Gis enseña cómo identificar números primos y compuestos. Los números primos son aquellos con solo dos divisores, el 1 y el número en sí mismo, como el 2 y el 5. Los números compuestos tienen tres o más divisores y se pueden expresar como el producto de números primos, como el 14 (2 x 7) o el 18 (2 x 3 x 3). Se destaca que el número 0 tiene infinitos divisores y el 1 solo tiene uno, por lo que ninguno de los dos es primo ni compuesto. El video invita a suscriptores a aprender más sobre el método de criba de Eratóstenes en la próxima clase.

Takeaways

• 📘 Los números primos son aquellos que solo tienen dos divisores: el número 1 y el número en sí mismo.
• 🔢 Ejemplos de números primos mencionados en el guion son 2, 3, 5, 7, 11, 13, 17, 19 y se indica que hay infinitos números primos.
• 📋 Los números compuestos son aquellos que tienen tres o más divisores y siempre se pueden escribir como el producto de números primos.
• 👉 El número 0 no es primo ni compuesto ya que tiene infinitos divisores.
• 👉 El número 1 tampoco es primo ni compuesto, ya que solo tiene un divisor, que es él mismo.
• 🔍 Los divisores de un número se encuentran comenzando por 1 y terminando en el número mismo.
• 📝 Se mencionó la importancia de anotar en un cuaderno las definiciones y ejemplos para no olvidarlas.
• 🎓 El guion ofrece un resumen de cómo encontrar los divisores de un número natural utilizando métodos convencionales y prácticos.
• 📚 Se invitó a los espectadores a suscribirse al canal de Gis y a dejar un pulgar arriba si les gustó el contenido.
• 🔍 Se mencionó que en una próxima clase se explicará el 'criadero de eratóstenes', un método para determinar qué números son primos hasta el 100.
• 📹 El guion es parte de una serie de clases en el canal de Gis, enfocándose en temas de matemáticas como números primos y compuestos.

Q & A

¿Qué es un número primo?

Un número primo es un número natural mayor que 1 que tiene exactamente dos divisores distintos: 1 y sí mismo.

¿Cuáles son los divisores del número 24?

Los divisores del número 24 son 1, 2, 3, 4, 6, 8, 12 y 24.

¿Qué números se mencionan como ejemplos de números primos en el guión?

En el guión se mencionan los números 2, 3, 5, 7, 11, 13, 17 y 19 como ejemplos de números primos.

¿Qué es un número compuesto?

Un número compuesto es un número natural mayor que 1 que tiene más de dos divisores distintos.

¿Cómo se puede escribir el número 14 como un producto de números primos?

El número 14 se puede escribir como el producto de los números primos 2 y 7, es decir, 14 = 2 × 7.

¿Por qué el número 0 no es considerado ni primo ni compuesto?

El número 0 tiene infinitos divisores y por lo tanto no cumple con la definición de un número primo ni de un compuesto.

¿Cuál es el único divisor del número 1?

El número 1 tiene un solo divisor, que es él mismo, por lo que no es considerado primo ni compuesto.

¿Cómo se puede verificar si un número es primo?

Para verificar si un número es primo, se deben comprobar que no pueda dividirse exactamente por ningún número natural mayor que 1 y distinto de sí mismo.

¿Por qué los números primos son importantes en matemáticas?

Los números primos son fundamentales en matemáticas porque forman la base de la factorización en números primos y son esenciales en el estudio de la teoría de números.

¿Qué es el método de criba de Eratóstenes y para qué se usa?

El método de criba de Eratóstenes es una técnica para encontrar todos los números primos menores que un número dado, y se usa comúnmente para generar listas de números primos.

Outlines

00:00

📚 Introducción a números primos y compuestos

El primer párrafo introduce el tema de la clase, enfocándose en la importancia de conocer los divisores de los números naturales. Se menciona que todos los números tienen divisores y se cuestiona si todos los números tienen el mismo número de divisores. Se invita al espectador a aprender sobre números primos y compuestos. Además, se pide a los espectadores que se suscriban al canal y den like si encuentran útil el contenido. Se da un ejemplo práctico de cómo encontrar los divisores de un número, utilizando el número 24 como referencia, y se comparan los divisores de otros números para destacar la diferencia entre los números que tienen dos divisores y aquellos que tienen más.

05:06

🔍 Características de números primos y compuestos

En el segundo párrafo, se profundiza en la definición de números primos y compuestos. Se aclara que los números primos son aquellos que solo tienen dos divisores: el número uno y el número en sí mismo. Se presentan ejemplos de números primos y se enfatiza que hay una infinitud de números primos. Por otro lado, los números compuestos son aquellos que tienen tres o más divisores y siempre se pueden expresar como el producto de números primos. Se da un ejemplo de cómo se puede descomponer un número compuesto en su forma de producto de números primos, utilizando el número 14 como ejemplo. Además, se plantea una reflexión sobre el número cero y el número uno, preguntándoles al espectador qué tipo de número consideran que son, ya que no se ajustan a la definición de números primos ni compuestos.

10:08

🚫 Explicación especial de números cero y uno

El tercer párrafo concluye la clase con una discusión especial sobre el número cero y el número uno. Se explica que el número cero tiene infinitos divisores, lo que lo coloca en una categoría aparte, ya que no se ajusta a la definición de números primos ni compuestos. Por otro lado, el número uno solo tiene un divisor, que es él mismo, y por lo tanto, tampoco es considerado un número primo ni compuesto. El instructor anima a los espectadores a anotar esta información en sus cuadernos y a no caer en engaños al tratar con estos números especiales. Finalmente, se invita a los espectadores a la próxima clase, donde se explicará un método práctico para determinar qué números son primos, utilizando el ejemplo del número 100 y se menciona el 'tamiz de Eratóstenes' como herramienta para este propósito.

Keywords

💡Divisores

Los divisores son números que pueden dividir exactamente otra cantidad sin dejar un resto. En el video, se utiliza el concepto de divisores para identificar números primos y compuestos, mostrándolos como 1, 2, 3, etc., hasta el número en sí mismo, como se ve en los ejemplos de los números 24, 18, 10 y 49.

💡Números primos

Los números primos son aquellos que solo tienen dos divisores distintos: 1 y el número en sí. El video destaca que los números 2 y 5 son primos y proporciona una lista de números primos hasta el 20, subrayando que hay una infinitud de números primos.

💡Números compuestos

Los números compuestos son aquellos que tienen más de dos divisores. El video explica que estos números siempre se pueden escribir como el producto de números primos, como en el caso del 14 que se puede expresar como 2 x 7, o el 18 como 2 x 3 x 3.

💡Método convencional

El método convencional es una técnica utilizada para encontrar los divisores de un número. Aunque no se describe detalladamente en el video, se menciona como una de las maneras de calcular los divisores, junto con el método práctico.

💡Método práctico

El método práctico es otra forma de encontrar los divisores de un número, que se menciona brevemente en el video como una alternativa al método convencional. Aunque no se detalla, se sugiere que se profundizará en una clase futura.

💡Criba de Eratóstenes

La criba de Eratóstenes es un algoritmo para encontrar todos los números primos menores que un número dado. El video termina con una referencia a esta técnica, prometiendo una clase futura donde se explicará cómo funciona.

💡Factorización en números primos

La factorización en números primos es el proceso de expresar un número compuesto como un producto de números primos. El video ilustra este concepto con ejemplos de números como el 14 y el 18, que se descomponen en productos de números primos.

💡Número cero

El número cero se menciona en el video como un caso especial que no es ni primo ni compuesto, ya que tiene infinitos divisores. Se destaca que cualquier número puede dividir a cero sin dejar un resto.

💡Número uno

El número uno también se considera un caso especial en el video, ya que solo tiene un divisor, que es él mismo. Por esta razón, no se clasifica como número primo ni compuesto.

💡Infinidad de números primos

El video enfatiza la infinidad de números primos, utilizando el símbolo de puntos suspensivos para representar que no hay un límite en la cantidad de números primos que existen en la matemática.

Highlights

Introduction to the concept of divisors and the special names for numbers with only two, three, or four divisors.

Invitation to subscribe to Gis's channel and engage with the content through likes.

Explanation of how to find the divisors of a natural number, using both conventional and practical methods.

Listing the divisors of the number 24 and explaining the significance of the number of divisors.

Demonstration of the divisors for various numbers, such as 18, 10, 49, and 5, to illustrate the concept of prime and composite numbers.

Definition of prime numbers as those with exactly two divisors: 1 and the number itself.

Identification of the numbers 2 and 5 as prime numbers based on their divisors.

Clarification that prime numbers have an infinite quantity, with examples provided up to the number 20.

Definition of composite numbers as those with more than two divisors and their ability to be expressed as a product of prime numbers.

Examples of composite numbers and their prime factorization, such as 14 being 2 x 7.

Explanation of the special cases of the numbers 0 and 1, which are neither prime nor composite due to their unique divisor properties.

Discussion on the divisors of zero, which are infinite, making zero neither prime nor composite.

Clarification that the number 1 has only one divisor and is therefore also neither prime nor composite.

Teaser for the next class, where a practical method for determining prime numbers will be introduced.

Mention of the Eratosthenes sieve as a method to be explained in the next class.

Encouragement for viewers to subscribe to Gis's channel and engage with the content for further learning.

Transcripts

Hi guys, you learned in previous classes how to find the divisors of certain

numbers, didn't you and do all numbers have several divisors and does the one that

only has two, three or four receive a special name? So to

find out this answer, I invite you to watch this class, let's go?

So welcome to another class on my channel, I'm Gis and in this class you will learn

what prime numbers are and what composite numbers are. But before starting the explanation, I

want to ask for those two things that you should already know what they are, don't subscribe to Gis's channel,

click on the little star here in the corner if you are not yet subscribed and leave a thumbs up

for me ok ? So before talking about what prime numbers are and what composite numbers are,

let's first remember how to find the divisors of a natural number, okay, I

'll do it here, so the divisors of the number 24, you've already seen in previous classes that we have

the methods of calculate the divisors of finding these divisors be it by the conventional method

or by the practical method right so here what are the divisors the number 24 let's create here

then the list of these divisors so I have, as we know that 1 is the divisor of all

numbers and itself so if I start if I want to find out the divisors of 24 here I know that I

will start at number one and I know that this sequence of mine goes up to the number 24 ok so here the

divisors of the number 24 are 1 2 3 4 6 8 12 and 24, if you want more detailed explanations on

how to find it using this conventional method or through the practical method, I will leave the indication

here for you and you can also find it in the link in the description ok because our objective

In this class, it is not about calculating the divisors, finding these divisors, but using these divisors

to try to understand what are prime numbers and what are composite numbers, okay? The divisors of 18

so let's list here the divisors of 18 I know it starts at one so it's 1 o 2 o 3 o 6 o 9 and 18 are

the divisors of the number 10, the divisors of the number 10 are 1 o 2 o 5 and 10, look here, now let's

do the divisors of the number 49, which are 1, 7 and 49, remember, it starts at 1 and ends with the number itself,

the divisors of five people, what are the numbers that I can divide five, okay, so remember well

if I put five here, what is the number that I put there inside the key that I can

divide, 5 is the number one that I can put there so it is one and what is the other number that I

can put if I divide five by two it will give 2.5, there will be a broken number left over, so

it can't be five by three, it won't be 5 by 4 either, so the only number that will

now divide the 5 that I can put in here is five itself, so the The divisors of five are

1 and 5 and the divisors of two so I can perform a division here who can I put there in the

divisor that gives the exact division I can put 1 because I already know that 1 is the divisor of

all numbers, who else can I put two themselves, right? Because remember, it starts with 1 and

ends with the number itself, 1 and 2 are good, but since the class, the focus of the class is not finding

the divisors that I want, so making these divisors I want you to understand something

the divisors of 24 when I listed the divisors of 24 how many divisors does 24 have

one two three four five six seven eight divisors so 24 has eight divisors then 18 18

has one two three four five six divisors 6 divisors okay 10 has one two three four

divisors 10 has four divisors right now here 49, 49 we found three divisors

for 49 and five five only has two divisors both the number 5 and the number two has only

two divisors so guys, this is where I want you to notice something, we have numbers

that have several divisors, you know, it doesn't even have eight divisors, even 8, 6 and 4 worked, but it was a coincidence,

isn't it always the case? the agent 8, 6 and 4 and 3 divisors 2 divisors and there are numbers that

have two divisors and based on the number of divisors that have a certain number we

will classify them into prime numbers or we will classify them into composite numbers and So

you already know that they are prime numbers, what are composite numbers, how am I going to classify them?

Look carefully, prime numbers are those numbers that have only two divisors, two divisors, mark them

in your notebook, okay, so from the list that I made here, in this list here, the two numbers that I

managed to put that have two divisors are the number 5 and the number two, they have two divisors each,

so I can tell you guys that the number 5 and the number 2 are prime numbers because they

have two divisors, so repeating again that it's for you to mark it right there and never

forget again, okay, who they are these two divisors of prime numbers is one and the number itself is one

and the number itself, that's why we don't say that prime numbers have only two divisors,

1 and itself being the number itself, okay, but what about the other numbers that don't even

this one that has three, this one that has 4, 6, 8 I can find some that have 12 10 divisors

they are called composite numbers that's right three or more divisors they are

called composite numbers two divisors prime numbers so mark this Note in your

notebook so you don't forget and let's now list what the prime numbers are, shall we?

And then we already marked in our notebooks that they are prime numbers, what are composite numbers, we haven't marked them

yet, so take advantage of what I wrote here on the board too, look, prime numbers, so

prime numbers only have two divisors, the number 1 and itself, so here I brought some examples

of some prime numbers, okay, I put them here in a sequence, so the numbers that are prime that

we have, I did it up to twenty, okay, guys, then we'll continue in the next class, okay,

2 3 5, remember 2 and 5 that we saw in the previous example, which are prime because they have two

divisors, 7 11 13 17 19, all these numbers that I put here are considered

prime numbers because they only have two divisors and one thing you must remember is that the numbers

primes are infinite, so it doesn't end here at 19 I wanted to simplify it up to here, it's only good because we

're going to do more prime numbers in another class, so to represent that they are

infinite I'm going to put the three dots here, okay, showing here that they They are infinite,

okay? And what about the compound numbers you wrote in your notebook about compound numbers? are

numbers that have three, three or more divisors three or more, okay, and these composite numbers, people,

can always be written using a product of prime numbers, that's right, remember in the class

on decomposing numbers into prime factors, so we take a number compound we carry out

the decomposition and this compound number is written in the form of a product or prime numbers look at the

compound numbers that I brought here as an example, okay, it's not in a sequence 4 o 6 8 o 9 14 18 and

24 35 so all these numbers that I put here and remembering that they are infinite, they are called

compound numbers because they have three or more divisors, but what is this business there, Gis, that

you said that all compounds are inscribed, they can be written as a product of prime numbers,

for example let's take 14 here, take 14, how can I write this number 14

over a multiplication using actually a multiplication of prime numbers, 14 can

be written as twice 7, two is prime and 7 is prime, so 14 is compound because I can

write a form of two prime numbers 2 and 7 but that's the only reason it is compound not because if

I think of the number 14 it has how many divisors? 1 is the divisor of 14, 2, 7 and 14 itself has

four divisors, so they fit this definition here, if I take, for example, the number 18, will

I be able to write the number 18 using the product of prime numbers?

as the number 18 can be written as, 2 x 3 x 3 o, let's check 2 x 3= 6, 6 x 3= 18, or I can

write it like this o twice 3 squared because I have two factors here so I put in the

form of a power, right, so remember that here are some examples that I brought, okay, and

leave them marked in your notebook, otherwise I'll ask you again if you marked them, right? And

here is this question mark that I put here, what do I mean by this and the

number zero and the number one, I didn't mention them at all, what do you think that number

zero and the number one is? Is it prime or is it composite? Let's think about what the divisors of zero are,

people, the divisors of the number zero. Do you know what the divisors of the number zero are? Think here, what

number can I put here in the key that allows us to perform the division, can I divide zero

by one? I can. Can I divide zero by two? I can. And zero by 3? And zero by 4? I can also, because

if I divided zero by four it gives 0, and 0 x 4= 0 gives the division correctly, so it means that zero

has infinite divisors, so for that reason it is considered neither prime nor composite, okay, so

check it out that zero has infinite divisors, I'm going to ask if you're marking it in your notebook,

okay? zero has infinite divisors and the number one guys, what are the divisors of the number one, do you

already know? Think about it, the divisors of the number one are just the number one because at the moment I

perform the division, the only number I can divide here for one it is because

if I put two here 1 / 2 does not give an exact division so the number one has only one divisor

for that reason it is neither prime nor composite so zero and one are special cases write it down In

your notebook, zero and 1 are not primes or composites, they are not primes or composites, okay

because zero has infinite divisors and one has only one divisor, okay guys, so don't fall for any

tricks, right? So that was our class today about prime numbers and composite numbers and

you marked everything in your notebook, right people and I want to invite you to watch

another class in which I will explain a practical method, that a mathematician created this method for

us to determine which are the prime numbers, then I'm going to do more than 19 here, okay, we're going to

find even the prime numbers until we reach the number 100, but it doesn't stop there, let's continue

in this class, I'm going to do it until number 100, ok? So be sure to watch the next class in

which I'm going to make this eratosthenes sieve for you who are already curious to know the name, I'll end

up telling you, ok? So I'll wait for you in the next class, but don't forget to subscribe to

Gis' channel and if you liked it, explain it, leave me a thumbs up and see you next time, bye...