Prime Factorization

Learning Objectives Define prime and composite numbers. Identify prime numbers up to 100. Explain what prime factorization is. Use factor trees to find the prime factorization of a composite number. Use the division method to find the prime factorization of a composite number. Write prime factorization using exponents. Understand the uniqueness of prime factorization for any composite number. Ever wonder what the 'building blocks' of all numbers are? 🧱 Let's break down numbers into their most fundamental, indivisible parts! In this lesson, you'll discover how every composite number can be uniquely expressed as a product of prime numbers. This powerful tool helps us understand number relationships and is fundamental in many areas of mathematics, from s...

TermDefinitionExample Prime NumberA natural number greater than 1 that has no positive divisors other than 1 and itself.The numbers 2, 3, 5, 7, 11, 13 are prime numbers. Composite NumberA natural number greater than 1 that is not prime; it has at least one divisor other than 1 and itself.The numbers 4, 6, 8, 9, 10, 12 are composite numbers. FactorA number that divides another number exactly, leaving no remainder.The factors of 12 are 1, 2, 3, 4, 6, and 12. Prime FactorA factor of a number that is also a prime number.The prime factors of 12 are 2 and 3. Prime FactorizationThe process of finding the prime factors of a composite number and expressing the number as a product of these prime factors.The prime factorization of 12 is $2 \times 2 \times 3$ or $2^2 \times 3$. Factor TreeA diagram u...

Definition of a Prime Number A natural number $p > 1$ is prime if its only positive divisors are $1$ and $p$. This rule establishes the fundamental characteristic of prime numbers, which are the building blocks for prime factorization. Numbers like 2, 3, 5, 7 are prime because they can only be divided evenly by 1 and themselves. Fundamental Theorem of Arithmetic (Unique Factorization Theorem) Every integer greater than 1 is either a prime number itself or can be represented as a product of prime numbers, and this representation is unique, apart from the order of the factors. This powerful theorem guarantees that no matter how you factor a composite number, you will always end up with the exact same set of prime factors. For example, $12 = 2 \times 2 \times 3$ is the only...