Math: Real Numbers and Prime Factorization

Study Notes

Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed.
This theorem provides a way to factorize any positive integer into its prime factors.
Example: 12 = 2 × 2 × 3 (unique prime factorization)

LCM (Least Common Multiple)

The smallest number that is a multiple of two or more numbers.
Example: Find the LCM of 6 and 8.
- Multiples of 6: 6, 12, 18, 24, 30, ...
- Multiples of 8: 8, 16, 24, 32, 40, ...
- LCM = 24 (the first number that appears in both lists)

HCF (Highest Common Factor) or GCD (Greatest Common Divisor)

The largest number that divides two or more numbers without leaving a remainder.
Example: Find the HCF of 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- HCF = 6 (the largest number that appears in both lists)

Word Problems

Example 1: Tom and Alex are baking cookies. Tom needs 12 cookies for his recipe and Alex needs 18 cookies for hers. What is the minimum number of cookies they need to buy to satisfy both recipes?
- LCM of 12 and 18 = 36 (they need to buy at least 36 cookies)
Example 2: A bookshelf has 12 books on each shelf and a library has 18 shelves. What is the greatest number of books that can be removed from the bookshelf and still leave an equal number of books on each shelf?
- HCF of 12 and 18 = 6 (they can remove 6 books from each shelf)

Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order in which the prime numbers are listed.
This theorem provides a way to factorize any positive integer into its prime factors.
Example: 12 = 2 × 2 × 3 (unique prime factorization).

The smallest number that is a multiple of two or more numbers.
Example: Find the LCM of 6 and 8.
- Multiples of 6: 6, 12, 18, 24, 30,...
- Multiples of 8: 8, 16, 24, 32, 40,...
- LCM = 24 (the first number that appears in both lists).

The largest number that divides two or more numbers without leaving a remainder.
Example: Find the HCF of 12 and 18.
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 18: 1, 2, 3, 6, 9, 18
- HCF = 6 (the largest number that appears in both lists).

Example 1: Tom and Alex need to buy cookies in bulk.
- Tom needs 12 cookies for his recipe and Alex needs 18 cookies for hers.
- LCM of 12 and 18 = 36 (they need to buy at least 36 cookies).
Example 2: A bookshelf has 12 books on each shelf and a library has 18 shelves.
- HCF of 12 and 18 = 6 (they can remove 6 books from each shelf).

Every positive integer can be expressed as a product of prime numbers in a unique way, except for the order of the prime numbers.

The prime factorization of a positive integer is unique, meaning it cannot be expressed as a product of different prime numbers.
Every positive integer can be expressed as a product of prime numbers.

The Fundamental Theorem of Arithmetic provides a foundation for many results in number theory.
It has applications in mathematics, computer science, and cryptography.

Every positive integer can be expressed as a product of prime powers, where each prime power is of the form pn, where p is a prime number and n is a positive integer.
The prime factorization of a positive integer can be used to find its greatest common divisor (GCD) and least common multiple (LCM) with other integers.