Prime factorization

Here are the key points to learn when studying prime factorization:

1. Definition: Prime factorization is expressing a number as a product of its prime factors.

2. Primes: Understand what prime numbers are—numbers greater than 1 that have no divisors other than 1 and themselves.

3. Methods:

   • Divisional Method: Repeatedly divide the number by prime numbers starting from the smallest (2) until the result is 1.
   • Factor Tree: Break down the number into its factors and continue factoring until all factors are prime.

4. Uniqueness: Every number has a unique prime factorization (Fundamental Theorem of Arithmetic).

5. Applications: Prime factorization is useful in simplifying fractions, finding the greatest common divisor (GCD), least common multiple (LCM), and in various areas of number theory.

6. Tools: Familiarize yourself with tools for finding primes, such as the Sieve of Eratosthenes.

7. Practice: Solve various problems to enhance your understanding and speed in finding prime factorizations.

8. Connection to Exponents: Learn how to express the prime factorization using exponents for easier representation (e.g., $2^3 \times 3^2$).

By mastering these points, you will have a solid understanding of prime factorization and its relevance in mathematics.

When studying "Prime Factorization," focus on these key points:

1. Definition: Understand that prime factorization involves breaking down a composite number into its prime number constituents.

2. Prime Numbers: Familiarize yourself with prime numbers (numbers greater than 1 that have no divisors other than 1 and themselves).

3. Methods:

   • Division Method: Repeatedly divide the number by its smallest prime factor until you get to 1.
   • Factor Trees: Draw a tree diagram to visualize the factorization process, starting with the number and branching out to its prime factors.

4. Unique Factorization Theorem: Recognize that every integer greater than 1 has a unique prime factorization (up to the order of the factors).

5. Applications: Understand how prime factorization is used in simplifying fractions, finding the greatest common divisor (GCD), and least common multiple (LCM).

6. Practice: Regularly practice exercises to reinforce your understanding and speed in determining prime factors.

7. Tools: Use calculators or factorization algorithms for larger numbers to check your work or for more complex factorizations.

By mastering these points, you will have a solid foundation in prime factorization.

When studying "Common Divisibility Examples," focus on the following key points:

1. Definition of Divisibility: Understanding that a number $a$ is divisible by another number $b$ if $a$ can be divided by $b$ without leaving a remainder.

2. Common Divisors: Identifying numbers that can divide two or more integers evenly.

3. Prime Numbers and Divisibility: Recognizing the role of prime numbers in divisibility and how they can have only two divisors: 1 and themselves.

4. Greatest Common Divisor (GCD): Learning how to find the largest number that divides two or more numbers without a remainder.

5. Divisibility Rules: Familiarizing yourself with rules for common numbers (e.g., 2, 3, 5, 10) that indicate whether a number is divisible by them.

6. Applications: Understanding practical applications of divisibility in factoring, simplifying fractions, and solving equations.

7. Using Prime Factorization: Learning how to use prime factorization to determine common divisors.

By mastering these points, you will gain a solid understanding of common divisibility concepts.