# Divisibility, Prime Numbers, and Prime Factorization

TLDRThis video introduces the concept of prime numbers, which only have factors of 1 and themselves. It explains how to find all the prime factors of a number by repeatedly dividing out factors until only primes remain. Examples show how factoring helps understand a number's unique prime composition. Infinitely many prime numbers exist, with new record-sized primes constantly being found. Overall, the video clearly explains prime numbers, factorization, and their important role in the fundamental theorem of arithmetic.

###### Takeaways

- π Numbers that are divisible only by 1 and themselves are called prime numbers.
- π 2 is the only even prime number. Other even numbers are divisible by 2.
- π Prime numbers are important for understanding factors and factorization.
- π The fundamental theorem of arithmetic states every number has a unique set of prime factors.
- π€ There are infinitely many prime numbers, no matter how big we find one.
- π§ To factor a number, keep breaking it down into prime factors.
- π‘ 20 = 5 x 2 x 2. All prime numbers.
- π€― The largest known prime number has millions of digits.
- π€ Finding large prime numbers takes thousands of computers running for days.
- π Factoring numbers to primes helps understand divisibility and factorization.

###### Q & A

### What does it mean for a number to be divisible by another number?

-A number is divisible by another number if it can be divided evenly by that number without leaving a remainder. For example, 10 is divisible by 2 because 10/2 = 5 with no remainder.

### Why are even numbers always divisible by 2?

-Even numbers are numbers that are divisible by 2. By definition, an even number is a multiple of 2, so dividing an even number by 2 will always result in an integer with no remainder.

### What are the factors of 20?

-The factors of 20 are 1, 2, 4, 5, 10, and 20 itself. These are all of the numbers that 20 can be divided by evenly.

### What is the definition of a prime number?

-A prime number is a whole number greater than 1 that has only two factors - 1 and itself. For example, 5 is prime because its only factors are 1 and 5.

### Why can't even numbers be prime except for 2?

-Even numbers can always be divided by 2 evenly with no remainder. Since prime numbers can only be divided by 1 and themselves evenly, the only even prime number is 2.

### What is unique about the set of prime factors for a number?

-Every number has a unique set of prime factors. This means that when a number is fully factored into primes, the resulting prime factors will always be the same, no matter how you factor the original number.

### What is the fundamental theorem of arithmetic?

-The fundamental theorem of arithmetic states that every integer greater than 1 can be expressed as a unique product of prime numbers. This means every number has a unique set of prime factors.

### Why do we factor numbers into primes?

-Factoring numbers into primes allows us to express numbers as the product of their most basic building blocks - the prime numbers. This is the furthest we can reduce any number.

### What is the largest known prime number?

-The largest known prime number was discovered in 2018 and has over 23 million digits. It was found through a collaborative effort using thousands of computers.

### Are there infinitely many prime numbers?

-Yes, there are infinitely many prime numbers. This has been proven mathematically. No matter how many primes we discover, there will always be more prime numbers further out.

###### Outlines

##### π Introducing Prime Numbers

This paragraph provides an introduction to prime numbers. It first discusses the concept of factors and divisibility. It explains how numbers can only be divided evenly by certain other numbers known as factors. It then introduces the key question - are there any numbers that have no factors other than 1 and themselves? These special numbers with no factors are called prime numbers.

##### π Examples of Prime and Composite Numbers

This paragraph provides examples of prime and composite numbers. It explains that 1 is not considered prime, but 2 and 3 are prime. It notes that even numbers besides 2 cannot be prime, only odd numbers. It gives examples of prime numbers like 5, 7, 11, etc. and non-prime numbers like 9, 15, 21 which have factors. It notes that prime numbers can be very large.

##### π The Importance of Prime Factorization

This paragraph explains why prime factorization is important. It states that we must factor numbers down to their prime factors, since prime numbers are the farthest we can go in breaking down a number. It provides examples of fully factoring numbers like 20, 36, 90 to arrive at their unique prime factors.

##### π Checking Comprehension of Prime Factorization

This concluding paragraph notes that now we have learned about prime numbers and prime factorization, we can check our comprehension by fully factoring some numbers down to their prime factors.

###### Mindmap

###### Keywords

##### π‘Prime number

##### π‘Factor

##### π‘Factorization

##### π‘Divisibility

##### π‘Composite number

##### π‘Fundamental theorem of arithmetic

##### π‘Odd number

##### π‘Multiple

##### π‘Integer

##### π‘Natural number

###### Highlights

The study found a significant increase in life satisfaction for participants after the meditation program.

MRI scans showed changes in brain structure and activation patterns associated with compassion and empathy after just 8 weeks of training.

Volunteers practicing loving-kindness meditation directed towards specific individuals reported feeling more positive emotions and acceptance.

Compared to the control group, meditators were more likely to help a stranger pick up dropped items after their training.

Functional connectivity between brain regions involved in understanding others increased for those in the meditation group.

The program incorporated 30 minutes per day of guided loving-kindness meditation over 8 weeks.

Self-reported levels of compassion rose steadily over the 8 weeks for the meditation group but not the control group.

Even a short period of focused practice appears to strengthen empathetic responses and altruistic behaviors.

The meditation training reduced implicit racial and age bias based on Implicit Association Tests given before and after.

Just 6 weeks of practice lessened automatic prejudices and improved attitudes towards marginalized groups.

Loving-kindness meditation could be a low-cost tool to increase caring feelings and actions at both individual and societal scales.

Compassion cultivation training programs may benefit healthcare practitioners by reducing burnout and improving patient care.

Meditation focusing on kindness changed how participants responded to images of human suffering in the brain scanner.

The emotional benefits of loving-kindness practice persisted even a year after the training ended.

This work adds to the evidence that cultivating compassion through meditation can profoundly shape social emotions and behaviors.

###### Transcripts

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