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Unique Prime Factorization

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Unique Prime Factorization

Study Guide

Key Definition

The Fundamental Theorem of Arithmetic states that every natural number greater than 1 can be written as a product of prime numbers, and that up to rearrangement of the factors, this product is unique. This is called the prime factorization of the number.

Important Notes

Prime factorization is unique up to the order of the factors.
Use a factor tree to find prime factorizations.
Prime numbers only have themselves and 1 as factors.
Composite numbers have non-trivial prime factorizations.
Prime factorizations are useful for finding the GCF and LCM.

Mathematical Notation

$×$ denotes multiplication of factors

$^$ denotes exponentiation indicating repeated multiplication

$p$ denotes a prime number in factorization

$()$ denote grouping of factors

Remember to use proper notation when solving problems

Why It Works

Every composite number can be broken down into its prime factors because prime numbers are the building blocks of all numbers. Using a factor tree ensures all factors are prime.

Remember

To find the prime factorization, keep dividing by the smallest prime number until you reach 1.

Quick Reference

Prime Factorization:$36 = 2^2 × 3^2$

Understanding Unique Prime Factorization

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Video explanation of this concept

Beginner

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Beginner Explanation

Every number greater than 1 can be broken down into prime numbers multiplied together. For example, $12 = 2 × 2 × 3$. This unique breakdown is called its prime factorization.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the prime factorization of $36$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You have $72$ candies to distribute equally among friends. Here, 'greatest number of friends' means the largest divisor of $72$. Use prime factorization to determine this number.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Consider the number $1001$. Determine its prime factorization.

Challenge Quiz

Single Choice Quiz

Advanced

What is the prime factorization of $1386$?

Recap

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Review key concepts and takeaways