BAB 3 Menentukan Faktor Positif | Matematika Dasar | Alternatifa

Summary

The video script discusses the concept of positive factors in mathematics, particularly in relation to prime factorization. It explains how to determine the positive factors of a number, using 24 as an example to illustrate prime factorization. The script then explores the pattern and method to calculate the total number of positive factors for any given number, simplifying the process without the need for exhaustive listing. It further applies this method to numbers like 80 and 120, demonstrating how to quickly find their positive factors, thus providing a valuable insight into mathematical problem-solving.

Takeaways

• 🧮 The script discusses the concept of determining positive factors, particularly in the context of mathematical simulations.
• 📚 Positive factors are numbers that can divide a given number without leaving a remainder, and they are typically discussed in the context of prime factorization.
• 🔢 The example of the number 24 is used to explain prime factorization, which is 2^3 * 3, and it's noted that 24 can be divided by many numbers, not just primes.
• 🌟 The script highlights the importance of understanding the difference between prime factors and positive factors, with the latter including all integers that can divide the number completely.
• 📈 The presenter introduces a method to calculate the number of positive factors of a number by using the formula (a+1)(b+1) for a number expressed as 2^a * 3^b.
• 🔍 The script provides a step-by-step approach to finding the positive factors of a number, emphasizing the pattern and method rather than memorization.
• 💡 A key insight is that the pattern for determining the number of positive factors involves adding 1 to the exponents in the prime factorization and then multiplying these numbers together.
• 📊 The script uses the number 80 as another example to demonstrate the method, showing that there are 10 positive factors for 80, which aligns with the calculated pattern.
• 🔢 The presenter also explains how to apply this method to larger numbers, like 394, by first finding the prime factorization and then applying the pattern to determine the number of positive factors.
• 🎓 The script concludes with a summary that there is a hidden pattern in positive factors that can be used to quickly determine the number of factors without extensive calculation.

Q & A

What is the main topic discussed in the transcript?

The main topic discussed in the transcript is determining the positive factors of a number, with a focus on prime factorization and how it relates to finding all the positive factors of a given number.

What is the difference between a prime factor and a positive factor?

A prime factor is a prime number that divides a given number without leaving a remainder, while a positive factor is any positive integer that can divide the given number completely.

How is prime factorization used to find the positive factors of a number?

Prime factorization breaks down a number into its prime components. The positive factors can then be determined by considering all combinations of these prime factors raised to their respective powers, including zero.

Can you provide an example of how to find the positive factors of the number 24 using the information in the transcript?

The prime factorization of 24 is 2^3 × 3. The positive factors are found by considering all combinations of 2^a and 3^b where a can be 0, 1, 2, or 3, and b can be 0 or 1, resulting in 8 positive factors.

What is the significance of the number 8 in the context of the number 24's positive factors?

The number 8 signifies the total count of positive factors for the number 24, which is derived from the combination of powers of its prime factors (2 and 3).

How does the transcript suggest simplifying the process of finding positive factors for larger numbers?

The transcript suggests using a pattern where the powers in the prime factorization are incremented by one and then multiplied together to find the total number of positive factors without having to list them all.

What is the method to determine the number of positive factors for a number with a prime factorization of 2^a × 3^b?

To determine the number of positive factors, add 1 to each of the exponents a and b, and then multiply the results together.

Can the method discussed in the transcript be applied to any number to find its positive factors?

Yes, the method can be applied to any number by first determining its prime factorization and then using the pattern described to calculate the number of positive factors.

What is the prime factorization of the number 80 as mentioned in the transcript?

The prime factorization of 80 is 2^4 × 5^1, which means 80 is composed of the prime numbers 2 and 5, with 2 raised to the power of 4 and 5 to the power of 1.

How many positive factors does the number 80 have according to the transcript?

The number 80 has 10 positive factors, which is calculated by adding 1 to each of the exponents in its prime factorization (4+1 for the power of 2 and 1+1 for the power of 5) and then multiplying these numbers together (5 * 2 = 10).

Outlines

00:00

🔢 Understanding Positive Factors and Prime Factorization

The first paragraph introduces the concept of positive factors and prime factorization. It explains that positive factors are all the whole numbers that can divide a given number without leaving a remainder. The paragraph uses the number 24 as an example to demonstrate prime factorization, which is the process of breaking down a number into its prime components. The example shows that 24 can be factored into 2^3 * 3, meaning 24 is composed of the prime numbers 2 and 3. The paragraph also discusses how any whole number can be a factor of 24, not just primes, and that the number of factors can be determined by considering all possible combinations of these prime factors.

05:00

🔑 Factor Counting Using Prime Factorization

The second paragraph delves into the method of counting the number of positive factors of a number using its prime factorization. It uses the example of the number 24, which has been prime factorized into 2^3 * 3. The paragraph explains that to find the total number of factors, one must consider all possible combinations of the powers of the prime factors. For the prime factor 2, there are four possibilities (0, 1, 2, 3), and for the prime factor 3, there are two possibilities (0, 1). Multiplying these possibilities together (4 * 2 = 8) gives the total number of factors for the number 24. The paragraph also extends this concept to larger numbers, like 80, and shows how to apply the same method to determine the number of its positive factors.

10:03

🔍 Expanding the Factor Counting Method

The third paragraph expands on the factor counting method by considering more complex prime factorizations. It discusses how to handle larger numbers with multiple prime factors and higher powers. The paragraph uses the number 80 as an example, which has a prime factorization of 2^4 * 5. It explains that to find the number of factors, one must add one to each of the exponents in the prime factorization and then multiply these numbers together (5 * 2 = 10), resulting in 10 positive factors for 80. The paragraph also touches on the idea that this method can be applied without having to list all the factors, making it a more efficient way to determine the number of factors for any given number.

15:06

📚 Applying the Factor Counting Method to Various Numbers

The final paragraph applies the factor counting method to different numbers, demonstrating its versatility and ease of use. It uses the numbers 394 and 120 as examples, showing how to determine their prime factorizations and then calculate the number of positive factors by adding one to each exponent in the prime factorization and multiplying these numbers. For 394, the prime factorization is 2 * 7 * 13 * 17, leading to 16 positive factors. For 120, the prime factorization is 2^3 * 3^1 * 5^1, resulting in 16 positive factors as well. The paragraph concludes by emphasizing the hidden pattern within positive factors and how the method can be a useful tool for solving related mathematical problems.

Keywords

💡Positive Factors

Positive factors, in the context of the video, refer to the whole numbers that can divide a given number without leaving a remainder. The video script discusses how to determine the positive factors of a number, using examples like the number 24. The script explains that these factors are not just prime numbers but any integers that result in a whole number when the given number is divided by them. This concept is central to the video's theme of exploring number properties and factorization.

💡Prime Factorization

Prime factorization is the process of breaking down a number into its smallest prime factors, which are the prime numbers that multiply together to result in the original number. The video uses the number 24 as an example, showing that its prime factorization is 2^3 * 3. This concept is fundamental to understanding how numbers are built from prime components and is used to explain the relationship between prime factors and positive factors.

💡Divisibility

Divisibility is a key concept in the video, which refers to the ability of one number to be divided by another without leaving a remainder. The script explores various numbers and their divisibility by different integers, such as 24 being divisible by 1, 2, 3, 4, 6, 8, 12, and 24. This concept is integral to identifying positive factors and understanding the properties of numbers.

💡Insight

The term 'insight' in the video script suggests a new understanding or realization about a concept. The speaker mentions that the discussion about positive factors provides an 'insight' for the audience, indicating that the video aims to offer new perspectives or deeper comprehension on the topic of number theory and factorization.

💡Number Properties

Number properties are characteristics or behaviors that numbers exhibit, which the video discusses in relation to factors and divisibility. Understanding these properties helps in solving mathematical problems and grasping the underlying structure of numbers. The video uses the properties of numbers like 24 and 80 to illustrate how to determine their positive factors.

💡Combination

In the context of the video, 'combination' refers to the various ways in which factors can be combined to form the original number. The script explains that by considering different combinations of the powers of prime factors, one can determine all the positive factors of a number. This is illustrated through the example of the number 24, where the combinations of 2^a and 3^b are considered.

💡Pattern Recognition

Pattern recognition is a skill emphasized in the video, which involves identifying regularities or sequences in data. The script discusses how recognizing patterns in prime factorization can simplify the process of finding all positive factors of a number. This is demonstrated when the speaker outlines a pattern for determining the number of positive factors based on the powers of prime factors in the number's prime factorization.

💡Mathematical Simplification

Mathematical simplification is the process of making calculations or expressions more straightforward. The video script mentions simplification in the context of finding positive factors, where instead of listing all factors, a formula based on prime factorization is used to quickly determine the count of positive factors. This approach exemplifies the practical application of mathematical techniques to simplify complex problems.

💡Exponents

Exponents, or powers, are used in the video to describe the number of times a base number is multiplied by itself. In the context of prime factorization, exponents indicate how many times a prime number is included in the factorization of a composite number. The video uses exponents to explain the prime factorization of numbers like 24 and 80, and how they relate to the total count of positive factors.

💡Multiplication

Multiplication is a fundamental arithmetic operation that is central to the video's discussion on factorization and positive factors. The script uses multiplication to combine prime factors and to calculate the total number of positive factors. For instance, the speaker multiplies the possible exponents of prime factors to find the total count of factors for a given number.

Highlights

Exploring the concept of positive factors in the context of mathematical simulations.

Positive factors are typically derived from OSN, which is surprising when it suddenly appears in SNBT.

Understanding the difference between prime factorization and positive factors.

Prime factorization involves breaking down a number into a product of prime numbers.

Positive factors include all integers that can divide a given number without a remainder.

The example of the number 24 being factored into prime factors 2^3 * 3.

Discussing the various divisors of 24, including 1, 2, 3, 4, 6, 8, 12, and 24 itself.

Introducing the pattern of positive factors using the example of the number 24.

Explaining the pattern of positive factors as a combination of powers of prime numbers.

Using the formula a * b to calculate the total number of positive factors from prime factorization.

Applying the pattern to determine the positive factors of larger numbers, such as 60 and 80.

The prime factorization of 80 is 2^4 * 5^1, and the pattern is used to find the total positive factors.

Highlighting the importance of adding 1 to the powers in prime factorization when calculating positive factors.

Providing a method to determine the number of positive factors without exhaustive listing.

Applying the method to find the positive factors of 394, which is factored into 2 * 7 * 13 * 17.

Calculating the total number of positive factors for 394 using the pattern and resulting in 16.

Discussing the hidden pattern within positive factors and its application in factorization.

Using the pattern to determine the positive factors of 120, resulting in 16 factors.

Transcripts

menentukan faktor

positif ini pada fomo kayaknya ini tapi

enggak masalah fomonya ini ke arah

positif sih karena memang di soal

simulasi ada tipe seperti ini gitu ya

menentukan faktor positif dan biasanya

faktor Positif itu keluarnya di OSN

makanya gua kaget OSN Matematika itu ada

faktor positif tiba-tiba di snbt ada

gitu ya cara apa eh soal tentang

menentukan faktor positif but it's ok

Enggak apa-apa ini sebuah Insight baru

buat teman-teman semua nah sebelumnya

harus paham dulu sebenarnya apa sih

faktor positif gitu nah kita ingat

kembali waktu SD Kita pernah belajar

terkait faktorisasi prima faktorisasi

prima itu berarti

ee sebuah susunan perkalian ya bilangan

prima ya Yang mana Bil ee susunan

tersebut adalah eh menyusun sebuah

bilangan yang lebih besar gitu misalnya

gini ya ini

faktori

Sasi Prima gitu ya misalkan kita punya

angka

24 lah ya Katakanlah kita punya angka 24

ya ini contoh ya misalkan kita punya

angka

24 Nah dengan pohon faktor kita bisa

Tentukan bahwa di sini 4 / 2 ya bilangan

prima ya adalah

12 12 / 2 hasilnya 6 6 / 2 hasilnya 3

maka faktorisasi prima dari

24 itu adalah 2 * 2 * 2 * 3 atau 2^

3 Dik dengan 3 gitu Nah inilah

faktorisasi prima dari 24 nah tetapi ya

pada kenyataannya angka

24 ya ini e adalah sebuah bilangan bulat

yang bisa dibagi oleh sekumpulan

bilangan di luar ini gitu loh Apakah 24

hanya bisa dibagi 2 enggak juga 24 bisa

dibagi 3 Apakah 24 hanya bisa dibagi 2

dan 3 enggak juga 24 bisa dibagi 1 24

bisa di 6 24 bisa dibagi 8 24 bisa

dibagi 12

artinya sangat banyak gitu ya Ee

bilangan bulat positif yang dapat

membagi habis 24 ya kalau kita jabarkan

ya Ini karena 24 ini angkanya kecil ya

kan J kalau kita jabarkan

ya banyaknya angka yang bisa membagi 24

itu ya

1 du

3 4 ya terus 6

8 12 24 gitu kan Nah ini adalah

sekumpulan bilangan yang bisa membagi

habis 24 nah ini yang disebut sebagai

Apa faktor positif Ya ini yang kita

sebut

sebagai

faktor

positif gitu Jadi kalau isasi Prima itu

adalah bilangan prima ya atau perkalian

bilangan prima yang menyusun sebuah

bilangan bulat yang lebih besar gitu kan

Nah nanti bilangan prima ini ya bisa

membagi habis 24 gitu kan Kalau memang

dia dibutuhkan sebagai pembagi nah tapi

kalau faktor positif adalah semua

kemungkinan bilangan bulat yang bisa

membagi habis 24 gu ada berapa Ada 1 2 3

4 5 6 7 8 ternyata ada Del nah ini

Kebetulan aja angkanya lebih kecil ya

kan angkanya kecil maksudnya angkanya

kecil 24 jadi bisa kita petakkan gitu Ya

nah tapi bagaimana kalau misalkan

angkanya lebih besar Apakah ada

pola-pola tertentu yang bisa kita

gunakan gitu ya untuk e

mempermudah mencari masalah eh mencari

masalah mencari solusi dari sebuah

masalah ini gitu kan Nah kita bisa lihat

teman-teman hasil dari faktorisasi prima

24 adalah eh 2^ 3 di*

3 ya 2^ 3 eh Dik

3 yang mana ini kalau kita buat sebuah

kerangka gitu ya kita buat sebuah

kerangka bahwa ini 2^ a

di* 3^ p b Maksudnya apa B maksudnya

gini pangkat dari 2 ini itu adalah

segala

ee kemungkinan yang mana Nanti 2 pangkat

sekian

itu itu akan menghasilkan

bilangan-bilangan yang ada di sini

sekarang kita lihat 1 itu 2 pangat

berapa sih 2^ 0 2 itu adalah 2^ 1 4 itu

adalah 2^ 2 ya dan 8 itu adalah 2^ 3

gitu jadi pangkat dari 2 atau si a ini

itu ada empat kemungkinan 1 2 3 4 gitu

Jadi ada 0 1 2 dan 3 jadi ada empat

kemungkinan pangkat dari 2 gitu ya Nah

sekarang pangkat dari 3 atau si b-nya

ini ya Ada berapa banyak kemungkinan ee

B ya supaya 3 ini ya ya hasil 3 dari

pangkat b itu ada di sekumpulan bilangan

ini gitu itu nah sekarang kita lihat ada

3^ 0 ya 3^ 0 itu hasilnya S 1 terus 3^ 1

hasilnya 3 Lalu ada lagi enggak 3^ 2 9

enggak ada di sini ya enggak ada 3^ 3 27

enggak ada ya Jadi kemungkinan B yang

bisa kita taruh di sini itu adalah cuman

0 dan

1 gitu sehingga

sehingga apa ya sehingga kita kita bisa

menentukan banyaknya faktor positif dari

hasil Apa hasil perkalian banyaknya

kemungkinan pangkat ini gitu ya karena

di sini A kemungkinannya ada 4 dan b

kemungkinannya ada ada du maka Ya

tinggal kita kalikan aja ya

a * b berarti di sini 4 * 2 berapa 8 ya

sama ini ada del8 hasil perkalian dari

pangkat juga ada 8 gitu ya ya sampai

sini bisa dipahami ya teman-teman ya Nah

angkanya

24 ya kita bisa petakan Berapa bilangan

yang EE bisa ya dibagi Eh bisa membagi

24 ya bisa membagi habis menjadi ee 24

menjadi bilangan bulat gitu nah angka

lain ada enggak ada misalkan kita pakai

angka berapa Katakanlah 60 lah ya kan

atau 80 deh agak agak ekstrem ya gua

gedein lagi misalnya 80 80 / 2 adalah 40

Agi 2 adalah 20 /i 2 adalah

10 bagi 2

adalah 5 gitu Nah berarti di sini

faktorisasi primanya adalah 2 p 4 * 5^ 1

stop Enggak ada udah itu doang gitu nah

Berarti untuk melihat

ee

banyaknya Apa faktor positif harus kita

apain Apakah harus kita giniin juga ya

sehingga kita harus menghafal gitu ya

Beberapa bukan kombinasi beberapa ee

banyaknya angka yang bisa kita gunakan

untuk membagi habis 80 Apakah harus kita

jabarkan juga seperti ini ternyata

enggak ya kita bisa melihat polanya

teman-teman Nah sekarang kita lihat ya

Ini 2^ a * 3^ b kalau kita

perhatikan ya pangkat 2 pada saat

difaktorisasi Prima itu adalah 3 gitu

kenapa di sini 3 karena kita cuma

menulis 2 * 2 * 2 gitu ya jadi duanya

ada ee 2nya pangkat 3 kita tidak menulis

satu pada faktorisasi Lia sedangkan pada

faktor positif ya kita e kita menuliskan

satu di sini sehingga apa pangkat dari 2

pada faktorisasi prima ya itu harus

ditambah 1 gitu ya harus ditambah 1 gitu

Jadi kalau kita tulis di sini adalah

pangkat pangkat dari 2 itu harus

ditambah 1 berarti ee apa

namanya 3 + 1 nah terus di sini

ee pangkat dari

ee

3 ya pangkat dari 3 itu 1 gitu kenapa ya

lagi-lagi tadi ya kita hanya menuliskan

tig di sini ya kita enggak enggak enggak

enggak menuliskan semua kemungkinan

bilangan bulat yang ada di sini gitu

berarti apa

ya ya ini juga kita tulis sebagai apa

apa pangkat dari apa 3 di sini kan 1

berarti harus ditambah 1 juga gitu nah

terus kalau misalkan nanti faktorisasi

primanya ada banyak lagi bilangan ke

belakangnya gimana ya polanya sama

pangkat pada faktorisasi prima ditambah

1 dikali pangkat angka setelahnya juga

ditambah dengan 1 nah

sehingga akan menghasilkan seperti ini 4

* 2 berapa 8 ya sama

seperti ini gitu jadi jadi tanpa kita

harus bisa ee menjabarkan dengan hafalan

ya Ee angka-angka ini gimana gitu ya ya

Kita kan cuma ditanyain Berapa banyak

faktor positifnya y wis kita tinggal

lihat aja pola yang menggunakan ini tadi

Berarti kita hanya perlu menjumlahkan

pangkat pada faktorisasi prima berapa ya

pangkat du kan ee 4 kita tambah dengan

1 lalu pangkatnya 5 itu 1 tambah dengan

1 gitu jadi di sini 5 dikali berapa 1 +

1 2 berapa 5 * 2 10 berarti apa ada 10

faktor positif dari 80 atau ada 10

banyak ada banyaknya ada 10 bilangan

bulat yang bisa membagi habis

80 gitu

ada berapa misalkan ni kalau kita mau

jabarkan ya ya semoga semoga gua enggak

salah ya 80 bisa dibagi 1 tentu bisa

dibagi 2 ya bisa dibagi 3 enggak mungkin

bisa dibagi 4 Yes bisa dibagi 5 Yes bisa

dibagi 5 hasilnya e

berapa ya 16 Ya tentu juga ba bisa

dibagi berapa nih

8 bisa

dibagi 16 juga ya Atau bisa dibagi 10 ya

bisa dibagi 10 bisa dibagi hmm berapa ya

bisa dibagi 2 eh bisa dibagi 16 bisa

dibagi 20 bisa dibagi 40 bisa dibagi 80

ada berapa nih 1 2 3 4 5 6 7 8 9 10 nah

terbukti ya Ada 10 bilangan bulat yang

membagi habis 80 sesuai dengan faktor

positifnya yang kita cari dengan pola

ini gitu nah sekarang bagaimana dengan

soal yang di snbt yaitu bilangannya

394 ya ya Apakah kita harus juga ee

menjabarkan kayak gini ya Ya gua rasa

enggak perlu yang penting kita bisa

menentukan apa

ee faktorisasi Prima Ya karena

394 ya 394 itu kan kalau dibagi 2

hasilnya

1547 Ya dibagi 2 ya

1547 ya lalu kemudian Dibagi berapa lagi

dibagi 13 ya dibagi

13 hasilnya berapa

119 lalu dibagi dengan 17 eh dibagi

dengan 7 sor ini bagi 7 dulu Berarti ya

kebolak berarti

1547 dibag dengan eh berarti dibagi

dengan 7 ya berapa

221 ya dibagi 7 hasilnya 221 lalu

221 dibagi e

13 hasilnya adalah 7 17 berarti di sini

faktorisasi primanya adalah 2 *

7* 13 * 17 nah semua di sini pangkatnya

1 ya yes berarti menyelesaikannya gimana

ya ingat tadi pangkat dari masing-masing

bilangan pada faktorisasi prima ditambah

1 berarti 1 + 1 Dik 1 + 1 ya sor dikali

1 + 1 di* 1 + 1 2 * 2 * 2 * 2 berapa ada

16

gitu ya lanjut misalkan Bagaimana kalau

120

120 bagi 2 hasilnya 60 bagi 2 hasilnya

30 bagi 2 hasilnya 15 b/i 3 hasilnya 5

berarti di sini 2^ 3 * 3^ 1 1 * 5^ 1

berarti pangkat dari 2 itu 3 + 1 ya

pangkat dari 3 itu 1 + 1 pangkat dari 1

eh pangkat dari 5 itu tamb 1 berarti 4 *

2 * 2 berapa 16 Berarti ada 16 ee

bilangan bulat ya atau 16 faktor positif

ya yang dapat membagi habis 120 gitu

ya Jadi ada pola tersembunyi dalam ee

faktor positif ini ya Sekian dari gua

semoga

bermanfaat