Pre-Algebra
Table of Contents
Prime factorization is a cool way to break down numbers into their building blocks. It's like finding the DNA of numbers, showing us what prime factors make them up.
This skill comes in handy when we need to find the least common multiple (LCM) of numbers. The LCM helps solve real-world problems, like figuring out how many donuts and muffins a bakery needs to package efficiently.
Prime Factorization
Prime factorization of composite numbers
- Positive integers with factors other than 1 and itself (4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20)
- Expressing a composite number as a product of its prime factors
- Prime factors divide a given number without leaving a remainder (2, 3, 5, 7, 11)
- Steps to find the prime factorization:
- Divide the number by the smallest prime factor possible
- Continue dividing the resulting quotient by the smallest prime factor until the quotient becomes 1
- The prime factorization is the product of all the prime factors used in the division process
- Prime factorization of 24:
- $24 \div 2 = 12$
- $12 \div 2 = 6$
- $6 \div 2 = 3$
- $3 \div 3 = 1$
- Prime factorization: $2 \times 2 \times 2 \times 3$ or $2^3 \times 3$
- Prime factorization of 60:
- $60 \div 2 = 30$
- $30 \div 2 = 15$
- $15 \div 3 = 5$
- $5 \div 5 = 1$
- Prime factorization: $2 \times 2 \times 3 \times 5$ or $2^2 \times 3 \times 5$
Basic number concepts
- Factor: A number that divides evenly into another number without leaving a remainder
- Multiple: The result of multiplying a number by an integer
- Product: The result of multiplying two or more numbers together
- Divisor: A number that divides evenly into another number (also known as a factor)
- Factorization: The process of breaking down a number into its factors
Least Common Multiple
Least common multiple calculation
- Smallest positive integer divisible by all the given numbers
- Finding the LCM using prime factorization:
- Find the prime factorization of each number
- List all the prime factors, including repeated ones, from both prime factorizations
- Multiply these prime factors to get the LCM
- LCM of 12 and 18:
- Prime factorization of 12: $2^2 \times 3$
- Prime factorization of 18: $2 \times 3^2$
- List all prime factors: $2, 2, 3, 3$
- LCM: $2^2 \times 3^2 = 36$
- LCM of 8 and 12:
- Prime factorization of 8: $2^3$
- Prime factorization of 12: $2^2 \times 3$
- List all prime factors: $2, 2, 2, 3$
- LCM: $2^3 \times 3 = 24$
Applications of prime factorization
- Real-world problems involve finding the LCM or greatest common factor (GCF) of two or more numbers
- Bakery packaging donuts (boxes of 12) and muffins (boxes of 18):
- Find the LCM of 12 and 18 to determine the least number of donuts and muffins for an equal number of full boxes
- LCM of 12 and 18 is 36
- Package at least 36 donuts and 36 muffins for an equal number of full boxes
- School filling up classrooms (24 students each) and buses (36 students each):
- Find the LCM of 24 and 36 to determine the minimum number of students required to fill up both classrooms and buses completely
- Prime factorization of 24: $2^3 \times 3$
- Prime factorization of 36: $2^2 \times 3^2$
- List all prime factors: $2, 2, 2, 3, 3$
- LCM of 24 and 36: $2^3 \times 3^2 = 72$
- Minimum of 72 students needed to fill up both classrooms and buses completely
Key Terms to Review (17)
Factor: A factor is a number or expression that divides evenly into another number or expression without a remainder. Factors are fundamental building blocks used in various mathematical operations and concepts, including multiplication, division, prime factorization, and the distributive property.
Product: The product is the result of multiplying two or more numbers or quantities together. It represents the combined or cumulative effect of the factors involved in the multiplication operation.
Divisor: A divisor is a number that divides another number without leaving a remainder. It is a fundamental concept in mathematics, particularly in the operations of division and factorization, that is essential for understanding various mathematical topics.
Divisibility: Divisibility is the property of a number being evenly divisible by another number without a remainder. It is a fundamental concept in mathematics that underpins various topics, including dividing whole numbers, finding multiples and factors, and prime factorization.
Multiple: A multiple is a number that can be divided by another number without a remainder. Multiples are an important concept in mathematics, particularly in the context of dividing whole numbers and understanding prime factorization and the least common multiple.
Prime Factorization: Prime factorization is the process of expressing a number as a product of its prime factors. It involves breaking down a number into the smallest possible prime numbers that, when multiplied together, result in the original number. This concept is fundamental in understanding various mathematical operations and relationships, including finding multiples and factors, as well as adding and subtracting fractions with different denominators.
Least Common Multiple: The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers. It is a fundamental concept in mathematics that is closely related to finding factors and multiples of numbers, as well as understanding the relationships between fractions and decimals.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is an essential concept in number theory and is closely related to finding multiples, prime factorization, and factoring polynomials.
Factor Tree: A factor tree is a visual representation used to find the prime factorization of a number. It involves breaking down a composite number into its prime factors in a hierarchical, tree-like structure.
Composite Numbers: Composite numbers are positive integers that have at least one positive divisor other than 1 or the number itself. They are the opposite of prime numbers, which have no positive divisors other than 1 and the number itself.
Prime Factors: Prime factors are the prime numbers that, when multiplied together, equal a given number. Prime numbers are positive integers greater than 1 that have no positive divisors other than 1 and themselves. Understanding prime factorization and the concept of prime factors is essential in topics such as finding the least common multiple of a set of numbers.
Exponent Notation: Exponent notation is a concise way of expressing repeated multiplication of a number or variable. It involves raising a base number to a power, which indicates how many times the base is multiplied by itself.
LCM: LCM, or Least Common Multiple, is a fundamental mathematical concept that represents the smallest positive integer that is divisible by two or more given integers without a remainder. It is a crucial tool in understanding and working with prime factorization and various mathematical operations involving multiple numbers.
GCF: GCF, or Greatest Common Factor, is a fundamental concept in mathematics that refers to the largest positive integer that divides two or more integers without a remainder. The GCF is crucial in understanding prime factorization and factoring polynomials, as it helps simplify expressions and find common factors among numbers or algebraic terms.
Factorization: Factorization is the process of expressing a number, polynomial, or algebraic expression as a product of its factors. It is a fundamental concept in mathematics that is essential for understanding prime factorization, the least common multiple, and operations with monomials.
Prime Numbers: Prime numbers are positive integers greater than 1 that have no positive divisors other than 1 and the number itself. They are the building blocks of all integers and play a crucial role in the concepts of prime factorization and least common multiple.
Fundamental Theorem of Arithmetic: The Fundamental Theorem of Arithmetic is a key principle in number theory that states that every positive integer greater than 1 can be expressed as a unique product of prime numbers. This theorem forms the foundation for understanding prime factorization and the least common multiple of numbers.